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.

You do not need to have GNOME in order to run GTK apps - Pop_OS ships with GTK/QT libraries so all apps works as on Ubuntu.

Who wants to run multiple widget sets though? You lose any semblance of your system having a native look and feel.

GTK3, GTK4, and libadwaita applications do look native in COSMIC. In COSMIC Settings, navigate to Desktop > Appearance > Icons and toolkit theming. Or search "toolkit" and it will be a top result. In the context drawer that opens on the right, toggle "Apply current theme to GNOME apps". This will then allow the cosmic-settings-daemon to automatically generate CSS variables for GTK4 and libadwaita apps. If adw-gtk3 is installed, it will also apply those variables to GTK3 applications.

Qt4 and Qt5 applications are unsupported, but IgKh/CuteCosmic has a Qt6 Platform Theme that integrates with the cosmic theme system and applies that theming to Qt6 applications. Though it won't work in 24.04 because most of the KDE/Qt apps in this release are based on Qt5, and the version of Qt6 is too old.

I want to run applications written with both Qt and GTK.

I do not care much about "look and feel"

You say that, but you probably do care at least a little.

It benefits you when an application doesn't do surprising things. Even basic things like clicking in an edit control, it's better for you if that results in the same outcome across all apps (stuff like does it select all text? does it place the caret at the end?).

Even for command line apps, consistency is good (see https://pubs.opengroup.org/onlinepubs/9699919799/basedefs/V1...). Are options specified with a hyphen? Two hyphens? A slash? Are lists delimited with comma or semi-colon or something else?

I personally try to run Qt stuff only but a lot of stuff most people will run is either GTK (Firefox, thunderbird) or Qt (VLC, Krita, Libreoffice by default). COSMIC can theme GTK to match its style, and so can KDE to an extent