It is interesting to me that you didn't encounter problems with the rote geometric method before abstract algebra. I went to a so-so public school in the American South, so I don't think I exactly had a stellar introduction to math or even school/academics in general, but beginning in trigonometry, we had to structure our proofs with more words and 'connective tissue,' and less explicit axiomatization/symbolization. Calculus continued this trend, and by the time I took linear algebra (at a local university while I was still in high school), the proofs were exactly like the proofs I would write throughout college. But there were several 'stepping stones' away from the original geometric proof.

Though admittedly I only got an SB in Math, I think geometric proofs were an okay introduction to the idea of the proof and of logic. Of course they are not representative of professional math. There are many things we learn as an introduction which don't turn out to bear that much resemblance to the more advanced form. I don't write 5-paragraph essays with clear thesis sentences anymore either. But it helped me to learn to write that way as a way of clarifying that I was stating my arguments effectively.

Sometimes, for my homework, I tried to state very explicitly in my proofs which axiomata I was using, both as an intellectual exercise and to see, e.g., where I might have gotten my logic backwards or where I used the axiom of choice (which constantly surprised me). I wouldn't turn these overexplicit proofs, but it helped to clarify that I was doing things correctly. If anything, when I was in college, the trend was away from 'informal' language. When I took, e.g., Discrete Math, for more complicated proofs, we were encouraged to write out a DAG for the dependencies of the proofs/lemmata so that it was exceedingly clear what depended on what and that we had in fact proven what we set out to prove.