Representations are usually defined as group homomorphisms from a group to a space of linear transformations, but in fact they are perceived by mathematicians as modules over a group ring. If you don't internalize the concept of a module in this context, then indeed you will miss out on a lot of the "logical connections" and "mental flourish" that you are seeing from other students.

You can't get this sort of "clarity of thought" just from going back and forth through defintions and theorems. You need to do lots and lots of problems successfully (not just spend hours on problems without progress -- but actually gain the insights that the problems are intended to teach you). If you're stuck because you can't get through even a single problem, then you need some help getting unstuck. Talk to your professors and fellow students. When you see proofs written by your classmates, try to re-cast these proofs from both perspectives: representations as a group homomorphism, and representations as modules. Become fluent in both viewpoints so that you can translate back and forth between them without effort. Ask classmates and professors specifically to explain their thought process and how they arrive at the proofs in their minds, not just the written proof itself.

Representation theory is really hard. For some people it's the first hard subject in math that they encounter. (For other people, the first hard subject is differential geometry, or algebraic geometry, or functional analysis.) Learning hard topics in math requires more than just time and effort. You need time and effort, intelligently applied, with accompanying insights and intuition. Talking to people and asking the right questions is a far faster way to achieve insights and intuition than reading from a book, but asking the right questions is a skill unto itself. Even if you are talking to people and asking the right questions, it still takes time and effort, but not an absurdly superhuman amount. Also, if you've learned one of the hard subjects, then you kind of have an idea how to do it in general, which helps the second time around, and then the third time around, etc.

I don't think learning math is beyond you, but I do think you should reconsider your approach. Internalizing math comes from experience, practice, and usage, and can be attained more quickly by talking to people. Reading a book is something that you do only when there is no other choice.