Broadcom BC4313 wireless finicky behaviour

Actually NM (like connman) is just a bunch of connectivity scripts and wont do anything useful if there's no proper driver administering your hardware.
You don't need to uninstall or reinstall anything as you can just disable/enable either NM or connman as a service at will, as I've tried to explain here.

You're barking up the wrong tree, it's kernel hardware modules/drivers with non-free firmware you need .... a level more basic than the connectivity programs that cannot do anything if the hardware isn't handled properly.

  • I'd like to walk you through but I don't have the hardware nor can I emulate it, so sorry about that.

Looks like you're in for a crash course hardware recognition, proprietary firmware and kernel module loading. :scream:
No easy copy/paste commands, alas. Not that those are really easy ..... they tend to be a trap if you don't understand what they do.