I believe the reason why you do not see any activity in the Live Activity pane when you try to simulate an incoming call is your browser settings
CTI adapter works with HTTP or HTTPS protocol. This is controlled in scoping.
Can you check if you have enabled HTTPS for CTI. If you do this, you will need administrator permissions on your PC to install/start CTI client adapter.
But, in this case, the browser settings should not affect because the parent window (cloud for customer) and the Live Activity embedded window are both using HTTPS.
But, if you are using HTTP communication for CTI, then you will need to enable mixed content on your browser.
You can search on google for instructions to enable mixed content for your browser.
Once you enable this, you should see the events showing up in your Live Activity window.
In the screen shots you have attached, I see that you are using chrome browser, and there is a small "shield" icon that appears on the address bar on the right end of the URL. This is indicating that the browser is blocking mixed-content and obviously blocking the CTI events from reaching the Live Activity pane.