Sorry, but I don't think this is a good idea. I don't want to have a complicated CI.
Especially building C from sources. A minor GCC/autotools update and good luck figuring out what is wrong. Nah.

Also, you cannot really forget to update those files, because tests would be failing.

Okay, I'll reopen it once there is progress on taiki-e/install-action#199

No, I don't think we need any changes to CI at all. It's fine.

Are you sure? I help maintain x11rb, which uses a lot of pre-generated code. Having a check like this is invaluable for preventing the usual mistake of "forgot to run the build scripts again".

forgot to run the build scripts again

It would not happen, because tests would fail. We have almost 2000 tests. The chance you forget something is tiny.
Again, simplicity is king. The current CI is good enough.