The rhetoric in the series certainly points at the FP and Sorcerer as being impressive, but not necessarily being very *deep* in their practice. Refinement is, as you've sussed, meant to be the way this is distinguished. If a FP is looking to spend another couple refresh on spellcraft, the expectation is that the advancement path is FP -> Sorcerer -> Wizard, rather than FP -> FP with more Refinement. (There are alternative paths, of course, where the FP ability sets evolve into something more like Sponsored Magic, but I'm talking main sequence here.)
That said, you won't *break* things (much, at least) by allowing greater amounts of refinement, for the "lesser" practitioners, but you'll also muddy the waters of those distinctions. If muddying those waters is attractive to you, do it! Harry's speaking from the perspective of the White Council; even if he thinks they're a bunch of fuddy-duddies, that doesn't mean he hasn't been soaking in the rhetoric, and maybe the rhetoric is wrong. On the other hand, series reality so far points strongly (if not 100% consistently) at the wizards as at least one weight class above the other kinds of practitioners, so keeping those distinctions unmuddied may give you an experience closer to what you get in the books.