Verify diagnostics more broadly in integration tests

[333fred]

#10232 (comment)

There are some unit test files (such as CodeGenerationIntegrationTest that don't verify C# or razor diagnostics of the code they produce. We need to do this much more generally in all our integration tests.

[chsienki]

@jjonescz @333fred I'll assign you both this one, but I assume @jjonescz will get to it first due to @333fred upcoming OOF.

Let's try and get this done soon, we're making a ton of changes and want to make sure our test coverage is as good as it can be to catch regressions.

[jjonescz]

I'm looking into this now.