I installed the new plugin and since my VMs were nicely showing up in it, I went ahead and uninstalled cockpit as I was unable to access it anyway (don't know what happened but I kept getting a NET::ERR_CERT_INVALID error (Safari would exit/crash immediately, but Edge at least showed me the error) and no reboot and no update would help).
What I didn't expect, though, was that uninstalling cockpit would also remove KVM, rendering the KVM plugin non-functional... So I reinstalled the new plugin and my machines reappeared on the list, but I'm not able to start them. Clicking on State -> Start will very briefly flash up a "Loading ..." modal and that's it. Nothing happens. The VM will not start.
Update: Since the VMs were set to auto-start, I got them to start by restarting the server. Since then, I am also able to start and stop them via the KVM plugin tab.
One thing I noticed, though: the tab doesn't refresh/update automatically when a machine changes state. It would be great if it could do that, at least when the change of state was initiated in that tab.