Does the following quote, copied from https://wiki.archlinux.org/title/Network_Debugging#Tcpdump, relevant?
they can only see outbound packets the firewall passes through: [https://superuser.com/questions/925286/does-tcpdump-bypass-iptables]
Perhaps you should disable the firewall, or loosen it, while debugging.
Thanks for the hint, but it does not apply: (one of) the clients doesn't even have a firewall enabled and I still cannot see the packages. To me, it looks, like it doesn't even try to send the pings, because it maybe thinks, the target is not reachable anyways ...
From my experience, tcpdump connects to the interface and you will see all traffic regardless of firewall settings, given you have the permissions. In your case I'd first verify that layer 2 is working correctly (layer 2 is ethernet or wifi). So I'd use the utilities provided by "wpa_supplicant" or "iw" to see if the "hardware connection" is working as expected. If your wifi card didn't connect on layer 2 it has no reasons to configure layer 3 (IP, IPv6) and above. Regards, Uwe