You'd need "a website" to redirect things, but if you've fully encoded the URL in the lengthened version, that website need not have any server side components. It could serve things that are handled purely client side.

As a trivial example, consider a URL base64-ifier. You enter the URL, it spits out base64urlifier.example/?base64=[encoded URL] - this is trivially done in Javascript based on the inputs. To redirect, all you need to do is go to that URL, and the Javascript reads the query parameter, de-base64s it, and redirects you there. No need for anything server side.

If you designed the service like this (which you have more than enough entropy for in the lengthening side of things - encoding 200 bytes of URL in 5 bytes of shortened link is hard, encoding 200 bytes of URL in 4kb of URL is easy), you wouldn't have any server side components beyond "serving some HTML and Javascript." Put it in a static file host, use the free tier of Cloudflare, and you can scale basically infinitely without any actual server load (if your service is barely used, hits to the static host backend are cheap, and if it's being used heavily, it's always in cache so never hits the backend).

There's no reason every web service needs a webserver, database, and anti-bot services.