I run a public URL shorter and you are correct:

it is relatively simple and inexpensive to operate and there is no reason it can’t continue operating into the indefinite future.

I disagree with the op on one major point… They suggest that the proper way for a service like this to operate is with a public database of links…

However, when I think of all of the different ways, one might use a shorter, it seems obvious that many links would necessarily be sensitive and/or private.

Therefore, I consider myself to have a dual mandate: privacy of the links created, and a duty, that I have assumed, of running the service for decades.