UniFi APs can run by themselves without the controller unless features like guest portal is enabled (as UniFi controller also functions as a captive portal). Restarting the controller won't restart your APs.

Is the UniFi controller necessary?

To get the full benefits of the UniFi ecosystem, I recommend that you have an always-available controller, no matter how it's hosted. If you don't want to run the UniFi Controller software, you can setup a UniFi access point in standalone mode using the mobile app.

Can I use a UniFi AP on its own?

You can also add UniFi switches if you want to extend it those networks to other ports in your house. It's easy to fall down the rabbit hole and spend a whole bunch of money and time on it, but the short version of this long answer is that there's nothing wrong with just using the APs on their own.