It would probably be easier to name the distros that don't have have a graphical updater tool

I think nearly all popular ones that come with a graphical interface at all. There is a graphical updater by KDE called Discover that works on a lot of different distros.

What distro doesn’t? Beside a couple probably.

not so much a specific distro. I know KDE has its own graphical updater in the "discover" app store, COSMIC DE also has its own app store. personally I always update my apps through the terminal as its just faster and easier.

All the Debian and RHEL based ones I've run into have it.

So Debian, Ubuntu, PopOS!, Rhel, Fedora, CentOS for the ones I've seen lately

the mostly populars probably

If I'm not mistake CachyOs comes with Shelly preinstalled now. While it can be a bit flaky with some esoteric aur packages it works fine for most things: pacman, flatpak, aur, even AppImage but I have not tried that.

Arch-based

It's not recommended, but you can use any graphical updater you want, by installing packagekit-qt6. The reason it's not recommended, however, is that if something goes wrong, you'll have a harder time figuring it out without pacman's logs; also, it won't merge .pacnew files, increasing the chance that something goes wrong after a reboot. If you just wanna use a graphical updater for flatpak stuff, that'd be fine, but you'd still need to use pacman -Syu to update system packages.

Garuda has a good one

openSUSE Tumbleweed has a graphical installer you can use to install, remove, update applications, browse, search, etc. But you still need to run a command line 'zypper dup' to fully update the system.

Most of them offer some sort of GUI updater. Zorin, Bazzite, Cachy, off the top of my head. How intuitive are they? Not. How much user input do they require? In the case of Zorin and Cachy, quite a lot.