I remember doing that about 12 years ago when I took over a folk club web site that had a "Links" page. It was a long and tedious task testing all the links, checking to see if the sites looked current (if they even still existed) and especially trying to find someone to contact if the site had problems and you wanted to know if the club or organisation it represented was still there.

It would help if all web sites displayed a "webmaster" address somewhere, or if the site has a domain name of it's own, that the email address webmaster@example.com was valid and reached a real person. I currently host about 30 domains and (I hope) every one has a webmaster@... address that redirects to an appropriate email address, even if it's a customer's site that I don't maintain myself.

I had another problem with mine, which was that I gave it a proper domain name, which it didn't have before, but the old site had links from all over the place, so I had to find those and write to their webmaster's addresses (again if I could find them) and ask them to change their links. The old site had also been set up by a student on a college web server and remained after the student had left, so I had to find someone at the college to remove the old site (and they didn't comply with my request for a 301 "Permanently moved" redirect which would have been more useful)

It is an endless and thankless task, cleaning up the internet, mitigated only by the occasional encouraging responses from people who actually exist, do something you've asked them to and reply to say they've done it.

Keeping a site up to date is hard. I run several and the job is never done.