Even easier, create predefined ports in caddy and connect to those via ssh. Assign domains to them like proxy<1-1000>.example.com. Connect to one via ssh that is free. Done. That's 2 seconds once it is setup.
Even easier and cheaper if you do not have a "random server exposed to the internet", take one of the alternatives shown by op which allows to deploy a digital ocean or Hetzner vm automatically. Done, like a few seconds from the command line.
10$ a month for something like this is over priced but I guess you prove that it is a valid business.