If you are at the point that other people can use your software, then you should use v1. If you are not ready for v1, then you shouldn't be releasing to other people.
Because this comment, "The project is still in development, it might be stable enough for use in "real projects(tm)", but it might also still significantly change." That describes every project. Every project is always in development. Every project is stable until it isn't. And when it isn't, you bump the major number.
I think we can come up with a reason why bumping the version number each breaking change isn't an elegant solution either: You would end up with version numbers in the hundreds or thousands.