Add architecture case-study code to CI

[ericwindmill]

Page URL

docs.flutter.dev/app-architecture/case-study

Page source

No response

Describe the problem

Right now, the code snippets in the PR #11414 aren't under CI. To add them to CI and test them properly, each snippet would require a lot of context.
The code those snippets are copied from lives in flutter/samples. Given that the apps primary purpose is to back the documentation, I'd like to move the entire app from flutter/samples to this repository rather than fake the needed context in new snippets.

Expected fix

• Move the app from flutter/samples to a new subdir in flutter/website/examples
• Use docregions to add the snippets to CI

Additional context

No response

I would like to fix this problem.

• I will try and fix this problem on docs.flutter.dev.

[ericwindmill]

Belongs to #11374