Can you just use a Linode? It’s $5/month. I bought a Linode VM and set up a server and database on it. Main drawback is efficiency, idk how fast a linode works once you have users. But once you have users if you can make money off them, you can migrate to AWS.
I've heard a lot of good things about Linode over the years, they slipped my mind while thinking of the more manual options. I'll have to check them out!