Why will B propose to X first if their first choice is Y?

You’re absolutely right. I’m on my phone trying to imagine the scenarios in my head and trying to find the simplest possible example.

Maybe 4 participants is too few to clearly see what happens.

(A,X),(B,Y) as the proposals is fine for the point being made. The issue is if we reverse which group is the proposer, we still end up with the same matching, because (A,X) strictly prefer each other.

The minimal example with 4 participants is if each person P has unique preferences and P's 1st choice has P as 2nd choice. So in this case just flip X:

A:XY B:YX X:BA Y:AB

If A,B propose you get AX,BY. If X,Y propose you get BX,AY. Both are stable because the proposers are getting their first choice.

I find this algorithm to have a curious definition of the word "stable."

Not sure if you're referring to the actual definition. It's basically "no two participants from different pairs in the matching would prefer each other to their current matches." It seems an elegant definition to me.

This obviously doesn't factor in most of the complexity of romantic relationships: not knowing your own preferences, evolving preferences, incomplete/imperfect information in general, etc. Plus it assumes a global 1:1 matching across two categories, and people can of course be LGBT or choose to be single.

The big assumption here is that people are either proposers or receivers and never switch roles.

Which may be largely true for many people but it's definitely not a fixed thing!

When I was dating, I initiated with a lot of women, but the woman I wound up marrying messaged me first.

This assumption is built-in for Gale-Shapley algorithm.