Linux Mint - Looks nice, I spent only like 30-60 minuts on it so hard to say but I think that if I could bring Software app from fedora/opensuse gnome with flatpak and flathub amazing GUI I would spend some time on customazing the look and would be able to stay longer.

I mean Linux Mint has a software center that includes both Flatpak and "system packages" version

Bazaar is a nice gui app for flatpaks.

Have a look at Debian. If you set up Flatpak with Flathub (very easy) you can easily install all of the applications you're likely to want with the GNOME Software GUI. Debian is a community distribution.

In my cursory search of the web, it does not appear that gnome-software is available on mint, unless you fancy mucking around with PPAs and such. But there is good news!

Mint uses a desktop environment called Cinnamon, which is installable on arch-based systems and should be listed as an option in the CachyOS installer. And you can install the gnome-software package on top as well.

Since you have mentioned that you have an RX 9070–which is a relatively new card, you should stay away from LTS distros like Ubuntu LTS and OpenSUSE Leap. Those have older kernels and Mesa which will noticeably impact your graphics experience.

For GUI-based app installation: pretty much all desktop environments have an app for it (e.g. Discover on KDE Plasma). Use can use them install software packaged by your distro, or other sources such as flatpak/Flathub. As mentioned by others, there are some independent storefronts such as Bazaar as well.

Mint has a software manager and you can also install Synaptic.

Gaming on Mint works fine, but it's based on Ubuntu LTS releases, so you won't have the latest kernel or mesa versions. If you're using an RX 9000 series GPU, you should probably pick a distro with a newer kernel and mesa version to get the best performance.

On Mint there is a Software Center or you can just install .deb files from wherever you download files.

You can use Octopi to install and manage software on Arch Linux distros like cachyOS

How to install Linux software using GUI?

On your GUI Desktop Environment, you use your GUI Application Launcher to start the GUI Terminal emulator. Then you simply type "apt install foo". Easy.

See, the GUI is not harder then the normal way.

Spend some time learning how to install software using the terminal. It'll be worth it, I promise.