If the main site doesn't get updated much, would it be better to make everything into static pages, that way, there are no security issues and no old software to maintain. The old (early 2000s) site was very well designed and easy to use so I expect something similar wouldn't be too hard to make.

As for the forums, we did have a completely fresh start back in the mid 2000s, and of course, all these years later, Invisionfree is no more. I can't see any reason why the current forum cannot be migrated to something else. I'm not too familiar with Xenforo (is that the fork of vBulletin?). A lot of people seem to like SMF these days; it is quite nice as an end user, but I've no idea if you can migrate a vBulletin database to it. Is phpBB still developed?