You can generate your links with Cloudflare functions and save the links with Cloudflare KV. You can also serve the game with Cloudflare pages.
[1] https://developers.cloudflare.com/pages/platform/functions/
[2] https://developers.cloudflare.com/workers/learning/how-kv-wo...
[3] https://pages.cloudflare.com
Cloudflare offers generous free tiers, so you shouldn't have to pay anything.
I find Cloudflare's ecosystem much, much easier than all of the alternatives for simple projects. It's fast, the UIs are nice, the docs are great, and everything is usually deployable straight through GitHub and simple configs.