Alright, after uninstalling Proton VPN I was able to get a pop-up asking for approval of the VPN connection by Netbird. Is that normal behaviour? Shouldn't I be able to have multiple VPNs installed, although just running one at a time?
After installing Proton VPN (which gives the pop-up even though another VPN is already installed) I am able to have both VPNs installed. I can disconnect the one and connect with the other, and vice versa.
TL;DR: FIXED by uninstalling Proton VPN, connecting Netbird, then re-installing Proton VPN.