Hi Liolio -
I can relate your breathing symmetry vs need for air every 2 strokes. About a year or two into my re-learning to swim freestyle as an adult, I FORCED myself to breathe on my weak side because I thought that symmetry would look better, even out my stroke, improve my posture in the water and overall make me a better swimmer.
It did all that. It was ugly, uncomfortable and super-un-natural for a few months, but eventually it clicked into place. I forced myself to breathe at least 100Y per session on my weak side, then on every 3rd stroke, and then it clicked. Most importantly, I learned that pursuing improvement required discomfort and that un-natural feeling. People swimming in water is not natural. Learning to breathe on my weak side was a huge confidence builder. Breathing bilaterally every 3rd stroke improved my stroke in many ways... but I wanted to swim faster.
I suspect that breathing on 2's feeds the bulk of the swimming population the right amount of oxygen for the performance required of an athlete to maintain a good workout in the aerobic zone.
Terry (TI founder) usually answers this question like this: that for any given distance he will strive to take the same number of breaths on each side. But he does NOT say he always breathes on 3's. This implies taking a number of strokes breathing on 2's on one side, then switching to breathing on 2's on the other side.
Practicing in outdoor pools in California, I tend to breathe away from the sun going down the pool, and coming back, I breathe on the other side to keep looking away from the sun.
After my cardiologist gave me a clean bill of health last year, I started to push my speeds even faster. Breathing on 2's allows me to go faster than bilateral breathing (3's). If running out of breath is the problem (the kind that would be solved if you could sit on the wall for 15 seconds) then getting more air is the answer.
Now, I swim longer distances breathing on 2's and when I start running out of air, I can take a couple breaths in a row (breathing on 1's, on one side and immediately on the next). That allows me to keep up a near-sprint pace for longer and dig into the anaerobic zone.
It may seem like all the cool kids are breathing on 2's but that's a choice forced by fuel requirements. If you really want to argue that symmetry is irrelevant, look at the 50m sprinters. No fair removing them from the examples. Not one is breathing on 2's and they are REALLY symmetrical.