I have since been able to reproduce the issue, having the same problem but less setup:
Install the package
Set up the tunnel (Assign interface, configure a peer)
Set up a WAN-Rule to allow connections and do a test.
Done
After the pfsense is rebooted, my WAN-Gateway for ipv4 goes down (Services -> Gateways) and the firewall is not able to do an ipv4 based update-check or ipv4 based ping (using Select Interface automatically). Everything works using ipv6, I tested the ipv4 functionality using the context switch in Advanced setup -> Networking.
On the other hand, certain ipv4 services still work, namely:
name resolution using ipv4
pinging, if the WAN-interface is explicitly chosen
Disabling wireguard does not solve the issue, uninstalling and rebooting does though. Logs give no hint as of why this issue arises. Does anybody have an idea? Or shall I issue a github-issue to the corresponding wireguard-plugin?