Major version changes are often partly a signal about support
Recently (for some value of recently) I read The cargo cult of versioning (via). To simplify, one of the things that it advocates is shifting the major version number to being part of the package name:
Next, move the major version to part of the name of a package. "Rails 5.1.4" becomes "Rails-5 (1, 4)". By following Rich Hickey's suggestion above, we also sidestep the question of what the default version should be. There's just no way to refer to a package without its major version.
I am biased to see versioning as ultimately a social thing. Looking at major version number changes through that lens, I feel that one of their social uses is being overlooked by this suggested change.
Many projects take what is effectively the default approach of having a single unified home page, a single repository, and so on, not separate home pages, repositories, and so on for each major version. In these setups, incrementing the major version number of your project and advertising the shift is a strong social signal that past major versions are no longer being supported unless you say otherwise. If you move from mypackage version 1.2.3 to mypackage version 2.0.0, you're signalling not just an API change but also that by default there will not be a mypackage 1.2.4. In fact you're further signalling that people should not build (new) projects using mypackage version 1.2.3; if they do so anyway, they're going to extra work to fish an old tag out of your repository or an old tarball out of your 'archived past versions' download area.
As I see it, this works because people think of there being a single project 'mypackage', not two separate projects 'mypackage version 1' and 'mypackage version 2'. Since there is only one project, it's relatively clear that by default it only has one actively developed major version, the latest, since I don't think people expect most projects to be working in several directions at once.
(Where projects have diverged from this model the result has often been confusion and annoyance. The case I'm closest to is Python, where having both 'Python 2' and 'Python 3' has caused all sorts of problems. But I'm not sure that calling them 'Python-2' and 'Python-3' would have helped. Incompatible major version changes in major projects appear to be a problem, period.)
I initially thought that you'd have problems getting this clear signal in a world where things were 'mypackage-1' and 'mypackage-2' instead of 'mypackage version ...'. Now, though, I'm not so sure. You'd probably have to be more explicit on your project web pages, which might not be such a bad thing, but it might be clear enough to people in general.