The site should 410 removed pages insted of 404ing... 404 just means "not here anymore" and google will wait to remove it from their index until they have revisited the page a few more times over days/weeks.
A 410 on the other hand indicates that this page has been intentionally removed, and google tends to act quicker on those.
Alternatively, they could 301 the removed pages to the most relevant parent page if that would be a better user experience for those few who did navigate to them. (in my experience 301ed pages also get removed from the index faster than 404s)
Thanks, that's helpful information as well. I didn't know about the 410. We used to 301 some pages, but ran into duplicate content issues where Google would continually spider old 301'd URLs (even when there were no longer links pointing to the original bad URL).
A 410 on the other hand indicates that this page has been intentionally removed, and google tends to act quicker on those.