Time for a new hat, I think. If I understand your misunderstanding correctly, what you're missing is the issue of tempo predetermination, or the lack thereof. Live looping is kind of like a tap tempo affair, except that instead of delineating enough beats for tempo to be guessed, the looper delineates only the end of phrase. How can quantization be used when the tempo is set at 121.632? Unless, of course, that particular show, it happens to fall at 107.229?
Quantization requires a finer granularity of predetermination than phrase-level.