Problem/Motivation
Follow-up from #3086644: LegacyProject composer templates wrongly reference 8.x + fix test coverage
We don't have an automated test which ensures that the quickstart feature works with the drupal/recommended-project layout.
Proposed resolution
Extend the test(s) from #3086644: LegacyProject composer templates wrongly reference 8.x + fix test coverage to also perform the quick-start command.
Fix all the things that need fixing to allow this to happen.