Basically yes. Given how long the White Council has been around, it can safely be assumed that anyone with the Wizard template is either a member, or has been trained by someone who's had White Council training. Which makes sense given the setting. Any mortal with the potential to be a wizard is going to be drafted into the Council, or will have to hide from them, like Elaine does.
It's definitely a narrative restriction, rather than a mechanically-balancing one, and Sorcerers can take The Sight as well, if they have the Refresh. If a Sorcerer wanted to take additional Refinements, I'd just rule that first they have to take The Sight, to represent their growing power (Luccio comments on Harry's own Sight improving, or "coming in", when they're on Demonreach, so it's definitely partly a representation of that), after which, when they take their next additional Refinement, they change to the Wizard template. Does mean they'd need to be careful around Wardens if they don't want to be forced into joining the White Council, but that should be a concern for any powerful spellcaster.
I think that if a GM were going to allow non-Wizards to take as make Refinements as they like and not require a template change, then it's probably fair for those characters to be subject to compels for problems with technology, just like a Wizard.