If you create a new, default WordPress site in Local, are you able to create a Live link from that site?
When I look at that Local log, I’m seeing a few errors like what you mention in the screenshot. This one in particular makes me think that there’s some sort of issue with this computer not being able to create a secure connection over HTTPS to the Live Links infrastructure:
Error: Client network socket disconnected before secure TLS connection was established
Do you have any kind of antivirus installed that might be preventing applications from making external requests? Another option might be if this is a work computer, does the IT department install their own self-signed certificates?