Do I understand correctly that your proposed feature is like next-error-highlight,
but instead of highlighting the error locus, it highlights the error message?
Yes, although I don't think it's important to support a timer to turn off highlighting like next-error-highlight does since the error locus highlighting might get in the user's way in source buffers, but not in the next-error buffer.
Then I don't understand how it relates to compilation-highlight-regexp and
I don't see a reference to compilation-highlight-regexp or compilation-highlight-overlay in my patch or in this thread. Although I did use compilation-current-error to find the mark at the error message location, which I think is not appropriate since that would be compilation-specific, and I think it should be (point) instead. Is this what you mean?