"So, I think this hilarious, but link shorteners have to be stable, right?" Yes, a link shortener does need to be stable but luckily they are extremely simple

"So, I think this hilarious, but link shorteners have to be stable, right?" Yes, a link shortener does need to be stable but luckily they are extremely simple to operate. You should be able to run a very popular link shortener on a leftover 1U with very low associated bandwidth and power costs. It's not unrealistic for a person, or group, to commit to this indefinitely.