This is because I was under the impression that a formal review is the only way for me to get a raise.
This seems rather directly mistaken. If you feel you deserve a raise, communicate this directly, rather than artificially putting a formality in the way.
upon reviewing me, they either have to tell me I don't deserve what I think I deserve, or they have to pay me more
I think it's pretty safe to say this is a false dichotomy. A formal review could tell you all sorts of things, none of them having anything to do with money. Absent some kind of clause in an employment contract, they don't have to do anything.