I love both these ideas - the SiteImprove plugin with that nice high visibility listing, and the Castle content quality check popup (and the plugin architecture). I would advocate for the quality check to include content accessibility (alt tags etc.), if it doesn't already (to enhance our promise of being an accessibility-friendly CMS.)
Back in 2015, @polyester came up with the idea of an accessibility control panel, which would be a great addition - a rollup report for site content, with links to pages that need improvements in various categories. Could also turn on and off a setting making alt tags mandatory (and other stuff like that). He could elaborate. I like the idea of separating out a control panel for content accessibility from another one for content quality, maybe Castle already provides that.
Regarding SEO checks, is there anything useful that could be taken from the SEO plugin quintagroup.seoptimizer? For example, it can be a nice feature to have a separate field for description-as-shown-on-the-page vs. description-the-meta-tag. (And being SEO friendly is another thing we advertise for Plone.)