I'd vote for including reader specific checks. So for example there is a page in the calibre wiki that documents the various quirks of ADE. It would be good to check for those. For example, ADE refuses to handle anchors names that start with a number. Or top level <br> tags (though I believe that was fixed in recent releases). Then are the various quirks of webkit based renderers. A place they are documented is in the EPUB output plugin in calibre, which implements work arounds for all these quirks. And of course it needs to do checking of flow sizes. Even though the flow size restriction has been mitigated in recent Adobe SDKs, it's not unlimited nd I think it's good practice to encourage people to split up their HTML anyway.
But really the most important set of tests is rendering tests and those are impossible to automate, the best that could be done is identify certain constructs that are unlikely to render correctly and warn about them. Things like fixed whitespace containers, large (wide) tables and so on.
And as GeoffC said it would be very important to make the error messages actually intelligible. I'm no dunce and I have a hard time figuring out what epubcheck error messages mean. I can only imagine how cryptic it must be for people that are not as familiar with XHTML.