It appears that extraterm doesn't even start communication with the SSH Agent after the host key exchange handshake:
ssh logging: Verifying signature ...
ssh logging: Verified signature
ssh logging: Handshake completed
ssh logging: Outbound: Sending SERVICE_REQUEST (ssh-userauth)
ssh logging: Inbound: Received EXT_INFO
ssh logging: Inbound: Received SERVICE_ACCEPT (ssh-userauth)
Error: All configured authentication methods failed
[Connection closed. Press Enter to close this terminal.]ssh logging: Outbound: Sending DISCONNECT (11)
ssh logging: Socket ended
ssh logging: Socket closed
1Password does not appear to receive the agent request in the first place, while other terminals like the Windows Terminal and WindTerm work just fine.