This PR completely drops usage of the builder action in favor of new actions
introduced in home-assistant/builder#273. This results in faster builds with
better caching options and simple local builds using Docker BuildKit.
The image dependency chain currently still uses per-arch builds but once
docker-base and docker repositories start publishing multi-arch images, we can
simplify the action a bit further.
The idea to use composite actions comes from #162245 and this PR fully predates
it. There is minor difference that the files generated twice in per-arch builds
are now generated and archived by the init job.