Doesn't running faster and having that extra width have the effect of giving more physical space on the tape to the changes in magnetization? And therefore wider frequency range or more stable differences in amplitude?

Sort of like how SP (2 hour) on a VHS tape looks much better than LP (4 hour) or ELP (6 hour).

It does, but plenty of albums have been recorded on 4 track machines to 1/4" tape.

The real difference in quality is probably mostly in having a machine that's properly setup and calibrated, with a good transport to minimize flutter, etc.