Am I missing something, or would these essentially be implemented via DNS records? It's not clear to me that keeping the links in a database would be necessary at all (unless the DNS records are what you mean by "database")

DNS is only for resolving the host part. The path is not passing through a dns query.

In example.com/blah, the /blah part is interpreted by the host itself.

And apart from that I would indeed consider DNS records a database.