How can we test that a full node is fully operational?

As far as I understand, running a full node isn’t mandatory for AIT-2 but it’s recommended.
I’m wondering if I’ve set it up correctly. It seems that the registration page only tests validator nodes.

I received a suggestion from @matthias that we could run this on both machines and outputs should be the same:

curl 2> /dev/null | grep "aptos_state_sync_version{.*\"synced\"}" | awk '{print $2}'

but I get 0 on a machine with full node, the validator node has a peer number though.
Is this the correct way to test it?
Also I frequently encounter this error in full node logs.