don’t know a lot about fastcgi, but looking it up it seems like it also keeps a persistent process running to avoid execution overhead

More of less, the good thing about PHP + fastcgi is that it keeps a configure amount of minimum persistent processes (plus a dynamic number) to avoid execution overhead like you said... now there's an important detail, PHP processes are generic, meaning if you've 50 apps hosted in your server they can all be served with just one persistent process. No need to spin up 50 processes for each app like with node/go etc.

Supervisors shutting down containers/processes are good, but starting up is slow compared to the single PHP idle process ready to go for any app. :)