Compile `rust_doc_test` binaries in an action

[UebelAndre]

re: #1427 (comment)

I think it might be time to rethink the rust_doc_test rules. Right now the test compiles and performs tests as part of the test itself. Instead of doing this, the test binaries that perform doc tests should be compiled in an action and the binaries be the executable for running tests.

[UebelAndre]

rust-lang/rust#87022 is something that might enable this behavior.

[UebelAndre]

cc @scentini for awareness.

[UebelAndre]

Looks like some of the rfc is implemented. It might be worth revisiting this and trying to compile tests in the build action and run them in the test action. This would also cleanup a ton of code and build issues.