Check example showcase patches in CI

[rparrett]

What problem does this solve or what need does it fill?

When PRs are merged, the "example-showcase patches" can go stale and break tooling that relies on example-showcase.

For example, the automated screenshot diffing happening at https://thebevyflock.github.io/bevy-example-runner/. I also hit this recently in my own local screenshot diffing infrastructure.

Motivated by:

#12440
#11596
#11058

What solution would you like?

Add CI that checks to make sure the patches apply.

What alternative(s) have you considered?

Add CI-only rust code / APIs for doing the overrides that these patches are doing.

[BD103]

I'm thinking something like git apply path/to/foo.patch, and then checking if it exits with a non-zero code. This sounds quite easy to do in Github Actions.

Should this be included in the ci tool, or just Github Actions?