This problem was noticed on bug 2186 in comment #17, but we closed that bug without fixing this. The "New Connection" menu option, as well as all other menu options, are disabled after you have logged in.
Absolutely no idea what is wrong here. Time est pulled out of the nether region.
*** Bug 5331 has been marked as a duplicate of this bug. ***
I tried adding a system menu to vncviewer, but unfortunately that did not change anything. There is also something weird going on as the menu doesn't work even if you execute vncviewer directly. But it does work if you start vncviewer outside of its bundle.
*** Bug 5598 has been marked as a duplicate of this bug. ***
Note that since we have implemented bug 5504, there's a workaround now: It is possible to drag a .tlclient file to the ThinLinc Client icon in the Dock. This will start a new client instance, even if you are logged in. You can probably also use the "open" command or Applescript.
*** Bug 7693 has been marked as a duplicate of this bug. ***
Somewhere along the line things have gotten worse. macOS is now not even displaying a disabled menu for vncviewer, and won't even update the name, instead showing the previously focused application.
We can confirm this on macOS 11 here, but we have a user report that this happens on 10.14 and 10.15 as well.
We've now also tested 10.15.7 ourselves and confirmed the behaviour there.