URL shorteners offering vanity urls solved people not needing to deal with the hassle of purchasing a vanity domain via a registrar, setting up DNS records, and waiting for propagation.
URL shorteners came later, and solved the problem of people needing to deal with “more steps” in order to have short memorable urls. Achieving memorable urls in “less steps” is one of the major selling points of url shorteners.
However, url shorteners did not solve short memorable urls — DNS had already solved that. But, as you’ve pointed out, acquiring a domain and setting up DNS records takes more work than simply using a url shortener service.