There are some serious issues with this but I think there's a good idea in there. Do you want a critique, or is this just for the sake of argument?

Mostly for the sake of argument. Obviously its "rough draft" but it seems like it gives you most of what you want (I probably wouldn't allow the free upgrade, personally, but that seemed important to you) but remove most of the things that bug me. Feel free to redesign and repost it if you think it might work better (or at least be an equal alternative), but I probably won't have use for it for a while.

The basis varies depends on the magic. Mab's Unseelie Magic might be drawing on her realm or her mother or her own mantle or just nebulous greatness. The Merlin's ward magic might be getting it as a free extra for extreme expertise or burning a bit of his human soul or incurring some karmic debt.
Debt doesn't have to mean anything in-character because FP don't exist in-character. So the justification can be more or less anything.

Except that debt DOES mean something in character with Sponsored Magic (as I've pointed out before, just like Fate Points do mean something in character) and "self-sponsored" magic just borrowed the template seemingly without considering the narrative consequences. But I've pretty much gone over all of this before and really what it comes down to is that you think that agenda-related and other debt are the same thing, while I profusely disagree for reasons I've already stated.

And you can potentially justify the debt, sure, but Sponsored Magic has a single reason for all sponsored debt.
Since you mentioned the subject of Mab, she doesn't HAVE Unseelie Magic. She has "plot device-level magic" (per Our World). Whether it comes from Mab, the Mantle, Mother Winter or the abstract concept of Winter itself Unseelie Magic is something that only characters borrowing that power can have because its Sponsored Magic.
One of my problems then, is that "self-sponsored" magic borrowed this concept but didn't have a reason for it across the board other than that it was part of the Sponsored Magic template.

I don't know what this is supposed to mean, exactly.

Not really sure what got lost in translation. The FATE system is built on a narrative structure. When you read a book (Hero's Journey aside) typically the character is good at certain things (Skills) and most of the time she's going to succeed at what she's good at and fail at what she's not. Of course, its a game so we don't want the results to always be predetermined so we use the FUDGE dice, which only average 0. Typically when the hero DOES succeed or fail where she shouldn't its because of something in her background (Aspects). The hero usually gets beat up at the beginning (Compels) and fantastically outdoes all previous conceptions at the end (Invokes).
So... when you make a mechanic in FATE, you make it support the narrative. If it doesn't support the narrative (it doesn't fit or it doesn't have a compelling reason to exist) then its a bad mechanic (aside: This is why I don't think Soulfire is actually Sponsored Magic no matter what the RAW says).

For the sake of saving some argument, let me give a try at making what you want in a way that I would consider to be at least less problematic.

Excessive Specialization [-1]
For some reason, you've abandoned the traditional path of magical study. Maybe you never had a mentor. Maybe you're on the run from the White Council. Or maybe you're Mommy just always told you you were a unique little snowflake and you're going to be and to heck with the consequences.
Musts: You must have an Aspect which alludes to this ability (typically your High Concept)
Each time you take this, you must choose a field of magical study (eg: Ectomancy, Demonology). If you have taken both Ritual and Channeling in the same field you get a one-time refund of 1 refresh when you take this power. You may take this ability multiple times. It may not be applied to a category of magic (Warding, Summoning and Binding, Crafting).
* Just That Good. You gain +1 to Control OR Complexity and Power OR Strength and Frequency when dealing with magic in this field. This ability may only be taken once per field, not once per bonus type. This bonus stacks with itself and specializations. Control applies to both Thaumaturgy and Evocation magics.
* Combat Casting. You may cast relevant Thaumaturgic rituals at combat speed provided you draw all the necessary power in one exchange. You may take this ability once per field.
* True Ritual. You may use one of your rote slots for a relevant Thaumaturgic ritual. The Thaumaturgic rote automatically succeeds after a number of exchanges (min. 1) equal to the complexity of the spell minus your Control. All parameters (except target, unless the target is yourself) and including any bonuses from foci and the source of any extra shifts for complexity must be set when this power is taken. This ability may be taken once, plus one for any slots Rote Caster gives you. This ability may be combined with Combat Casting provided the rote takes one exchange.
* Rote Caster. You have gain a number of rote spell slots equal to your Lore that must be used for Rotes in the field of study.
* Easy Peasy. You may cast a single relevant rote for one less stress (min. 0) than it would normally take. You may take this ability once per rote.
* Of Course I Know That. You automatically succeed at any knowledge roll relevant to casting of spells in this field (eg: internal anatomy for Biomancy) provided the difficulty is less than your Lore.

I realize those abilities are kind of all over the map (and the refresh cost may be off because of it), but this basically reflects most of the things you'd want a specialist to be able to do generically without taking on any extra baggage.

EDIT: Added a couple clarifications including that bonus is +1 to Control for Thaumaturgy and Evocation OR a +1 to Complexity and Power.

I think I've been addressing what you said. You've been arguing that an agenda is a weakness, I've been arguing that agendas are the same as ordinary compellable aspects. Which aren't weaknesses.

Then we have a discord. (See what I did there?)

What I said about one opponent apples even more strongly to a group of opponents. Zone-wide evocations are great.

Haha. I think my example appled some effect after all. :-P

Also, stress recovers every scene. Not every session.
In my experience most fights, even epic ones, tend to end fairly quickly when evokers are involved because they hit hard and are fragile. Four exchanges is actually a fair while.

And that's kind of my point. If your epic fights are lasting four exchanges, you're probably throwing the wrong kind of encounters at people.

Yes they are. They're related, obviously, but the narrative doesn't have Fate Points in it.

The mechanics are subject to the narrative, not independent and not equal. That's what I'm saying.

And in my experience, the best way to make sure that a Power is balanced is by looking at it without considering the narrative. I'm pretty confident in this because I've designed a lot of Powers and watched other people design a lot of Powers. People who use narrative concepts to balance things tend to write bad Powers.

But most of them are. +2 to Lore when navigating the Nevernever IS a narrative balance. If it were just +2 for one refresh it would be too powerful, so its balanced by a narrative restriction. Being able to borrow debt without the narrative restriction of an agenda is too powerful. But really, the point here was the point above... if you make a Power that's based on another Power but it doesn't have the same narrative basis for one of its abilities it a bad power. In Sponsored Magic that debt is power borrowed from your sponsor. Where is the narrative basis for debt for "self-sponsored" magic? That's the most critical part of the question and the thing that no one seems to have answered.

The only reasoning I've seen for it is something to the effect of "drawing on inner reserves" but if that's the case that should just be a houserule that all players can do, although that's really what Fate Points are for in the first place. Fate Points are a resource management system. If there's no downside or restriction on taking on debt then you've basically wrecked the core mechanic of FATE.

Lawbreaker doesn't do anything to prevent people from hacking and slashing and adventuring. It's an obstacle for interesting character moments and backstories.

I'm not saying it prevents them. What I am saying is that for most characters this should never be an issue in the first place. If you don't like it don't use it. I just happen to think it represents the Dresdenverse consequences of lawbreaking pretty well. Break a Law? Become a little bit more of a monster.

...the young wizard is rational enough to not kill his mentor, just wound him, so he can kill him with a knife. That's rather anticlimactic, and he doesn't do it because it's what he would do, he does it because of a mechanical reason, and I'd rather avoid that.

I wouldn't call that anticlimactic, personally. In fact, incapacitating him, then walking over and killing the helpless enemy with a knife is a MUCH more personal and thus gripping narrative. It's also not JUST mechanical. Killing his mentor with magic is the sort of thing that would bring the White Council down on his head, so there's a narrative reason too. Since lawbreaking essentially requires a conscious choice (intent arguments aside), I happen to think the power represents it very well.

In some ways, I think this is only an issue in the first place because a lot of people come from the hack-and-slash adventuring mentality of D&D. Magic or no magic, murder is not something characters in a modern society should be taking lightly regardless.

Sometimes, it might just be right for the character to kill someone (or do something else lawbreakery). If you feel like you have to not do this, because your character would become an NPC for it, you will probably still kill the person, but you will look for a way to avoid the mechanical downside of it.

I feel like that's kind of the point though. It's like being a changeling and reaching the point in the story where you're like "I need more power to succeed." Knowing your going to become an NPC you're going to look for another way to do it, but in the end you might just decide that becoming an NPC to, say, preserve the natural order of the universe just might be the better option.
Likewise, with Lawbreaker, there are usually ways to avoid it. Killing them with not-magic, for instance. But maybe, instead, you decide to permanently change that character's mind; effectively neutering them or even making them one of the good guys. Maybe that's the sort of thing you think is worth becoming an NPC for. Of course, if you don't drop yourself to 1 refresh when designing your character this isn't immediately an issue. And the first time you break a law is a pretty major milestone so maybe the GM will poll the players at the end of the session/adventure and decide to raise the Refresh a point, just like maybe your GM doesn't feel like being a faerie doesn't have to make you a NPC. Of course, YMMV and maybe for your game its better not to have it, but personally I think it makes it a lot of sense and adds some mechanical "oomph" to consequences of breaking universal laws.

Keep in mind, as well, that as GM you are the god of your own little petty world. Realize your faerie told an outright lie? ("No, there isn't a faerie waiting to kill you at the end of that Way.") Tweak the world a tiny a bit and now they haven't. (The faerie directed them to a completely different path while they were still concerning themselves with all the dangers on their intended route.) It can be... problematic. And you want to be careful how often or how badly you do this, but if you avoid making simple declarative statements you can generally avoid lying outright.