Facebook repeatedly asks for a password

Not sure about this but when you deleted the apps did you reboot your device prior to adding them back again, also did you just reinstall the Facebook App first to see what happens before adding any others that probably use FB to sign in with.

The only other option is to try “Reset All Settings”, it doesn’t remove your data but you will have some reconfiguring and password entry work to do.

Do NOT; repeat DO NOT "Reset All Settings" you will have SOME reconfigureing and password entry" -- NOT SOME --- ALL your passwords; ALL your wifi settings and passwords!!!! NO way I would EVER do this.

It really depends on how many networks you connect to on a regular basis, most of us only probably have two or three main ones because we use cellular more often than not. If you also have iCloud Keychains enabled your other passwords for web sites etc should not be affected. Bluetooth should not really be an issue because syncing with your equipment is becoming easier and more straight forward (take the AirPods for example).

The only other option if all else fails to resolve this issue is "Erase All Content and Settings" which puts your device back to new (if you don't then restore a backup), but this requires far more effort to get your device setup the way you previously had it, so the lesser of the two evils is by far the "Reset All Settings" in my view.

It all depends on how bad the issue is and does that out weigh the effort required to do a little resetting up of your device again, lets face it, if I had an App constantly asking for a password at every turn I would consider refilling a few wifi passwords and a few app setups time well spent.

But in the end its for the OP to decide.