Add cargo-semver-checks to CI

The problem with this is that once it's triggered it triggers for every future CI job until you release the next version. Is there any good way to make this informational perhaps?

The problem with this is that once it's triggered it triggers for every future CI job until you release the next version.

I see, so otherwise, we'd have to bump the version number in the same PR that introduces the breaking change. I'm not sure that's necessarily a bad thing, though?

Releases happen eventually... with lots of code building up in main until then. There are probably ways to make the semver checks informational in a way that can work well. I haven't played with it though.