> It's complaining because I've modified README locally, which was also modified remotely. Every other reasonable version control system I've ever used will happily merge the upstream changes with my not-yet-committed local changes. But Git refuses. This is annoying.

No it isn't. Git is trying to be predictable.

> Like suppose I was working on one change when I realized that there's something else I should really fix first.

Branches are designed to resolve this problem. When you want to do something that takes a lot of time, you branch from the stable version. When you need to do something else, you branch from the stable version again.