I think one needs to be careful with this sort of argument. Do you completely understand an algebraic structure by writing a bare-bones implementation of it?

I think one needs to be careful with this sort of argument. Do you completely understand an algebraic structure by writing a bare-bones implementation of it? Absolutely not. Do you completely understand an algebraic structure by proving a few exercises in a book, or all of the exercises in a book? Also absolutely not. All either of these does is change the balance an infinitesimal amount between what you do know about the structure and the vast, staggering amount of things you don't know about the structure. Getting paralysed by that vast, staggering amount out there doesn't help anyone. Instead, take tiny nibbles and build confidence—and if your implementation in a programming language of an algebraic structure helps you to do the proofs, or even just helps you realise when the structure might be appropriate in a programming task, then real progress has been made, no matter how much you still don't know.