Linter action shouldn't install Taplo

[BD103]

Since we control the toolchain file and it doesn't look too complex, I wonder if we could accomplish the same without installing an additional tool. That could speed up the CI time.

On the other hand, we'd need to be careful to make it work on all platforms, so it probably makes sense to investigate this in a separate issue.

Originally posted by @TimJentzsch in #501 (comment). Also see my comment in #501 (comment):

I think this would be a good follow-up. We could probably use grep to capture the toolchain version with a few clever comments, like what Clippy does.