Refactor do_check routine to be more modular and address concerns about quiet flag

[matteius]

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The issue

Fixes #6228

The fix

Leveraged Google Gemini-1.5 large context to give a concise summary of image and proposed fix, given context of the complete code base. I think fed this into Claude manually and replaced the routine for do_check, and initiated test run.

The checklist

• [X] Associated issue
• [ ] A news fragment in the news/ directory to describe this fix with the extension .bugfix.rst, .feature.rst, .behavior.rst, .doc.rst. .vendor.rst. or .trivial.rst (this will appear in the release changelog). Use semantic line breaks and name the file after the issue number or the PR #.