I don't think there is all that much need to distinguish between user generated content, and content provided by the site itself when considering how to tag a question regarding a broken link.
For example if there is a broken link in a help center page we could use the tags:
I think this is the intuitive thing to do under these circumstances.
If a broken link appears elsewhere, there are plenty of other tags we could use to indicate that it is in system generated content. For example review for something in the review queue's or profile-page for something on a users' profile.
Therefore I suggest we make 404-not-found a synonym of broken-link and adjust the tag wiki and excerpt accordingly.
As Journeyman kindly pointed out in the comments we still have a legitimate use for 404-not-found, namely the custom 404 pages the network has. Therefore making the tag a synonym of broken-link is not the right solution.
We should rather narrow the scope of the 404-not-found to be specifically about the (custom) 404 pages the network serves, and re-tag questions about broken links to broken-link.
I'd imagine the excerpt for broken-link would become something along the lines of:
Use this tag when you want to report a broken link in content that can't be edited by users, or to discuss the effects of broken links within posts and comments across the network, what to do with them, and how we can prevent them from becoming broken.
And the excerpt for 404-not-found would become something along the lines of:
For discussing or reporting bugs with the (custom) 404-error pages of the network itself.
improvements are welcomed