No? They can run on cosmic, just like they can run on any other desktop. It's a linux distro, it does linux distro things and happens to be built on ubuntu. What part of this are you not getting? Remember when Ubuntu made Unity because they were pissed at the GNOME people? That's what System76 did, they got mad at GNOME because GNOME didn't like how much they were messing with GNOME
I'm not super familiar with the history of GNOME, KDE, or COSMIC and I've never used COSMIC and I haven't been able to see into their app store to see what apps are available.
The version of Chrome (for example) that Google distributes uses GTK which is GNOME, no? So I was wondering if System76 forked that and made a version that uses the COSMIC API.