I'm not very well-versed in economics, but would setting prices to a relatively non-changing BTC value not actually be a good thing to stabilize its value, if enough people were to do that?
What is the incentive to do so? BTC is expected to rise in value over time since it's a deflationary currency. So any sane businessperson should expect to be constantly decreasing the BTC price over time. It's a feature, not a bug.
Yes, but the same effect happens (in the opposite direction) with a normal inflationary currency. Every few months or years, prices change to reflect their new value. When prices are doubling or halving once a week, that starts to get problematic (see interbellum Germany, 2000s Zimbabwe).