Is the new UI broken or are app developers just not implementing it into their apps or what's wrong with the current situaltion?
Both, kinda. The new UI relies afaik on xdg-portals to get which apps are running in the background. Therefore only flatpaks should show up; but they should show up automatically, without any tweaks by the devs.
Also the UI only displays that an app is running in the background. It can't communicate any type of status information.