I guess you are not using the ssh utilities shipped with macOS, the system default ssh-agent daemon should creates a unix socket locates at dir /private/tmp/ and suffixed with Listeners.
System default ssh-agent daemon is launched by launchd and always starts up before Core Tunnel, so the SSH_AUTH_SOCK var is always ready when Core Tunnel starts.
If your ssh-agent not managed by launchd, then the SSH_AUTH_SOCK var may not set correctly for GUI applications.
It seems like it. But I am not aware I did install another ssh-agent. But I di install gnupg. I will check the gpg-agent setup. Maybe that explains it.
macOS shipped ssh-agent is launched by launchd utility, and by reading the rules in config file /System/Library/LaunchAgents/com.openssh.ssh-agent.plist, here is the content in my Mac:
According to the configuration, launchd will expose the SSH_AUTH_SOCK environment var on startup, so every GUI applications can read this var even itβs changed on each startup.