There are tools to check README files. Which would give suggestions on what needs to be fixed.
If you are using Sphinx anyway, would suggest going with a ReStructuredText rather than vanilla markdown.
In your Sphinx docs, can mix rst and md but the index file must be index.rst, not index.md
Then use this sphinx-external-toc-strict for structuring your Sphinx docs.
restview for real time editing of restructuredtext file