The most recent version of Steam on the Deck addresses this if you go to the "Non-Steam Game" page. It gives you the option to install Chrome and add it as a non-steam shortcut with one click. It also has a short note about how you can easily do it with any app in Desktop Mode, and that they're working on a way to do that within the Gaming Mode interface.

Not ideal, but they're working on it. Also, would prefer Firefox obv.

What store was broken?

Edit: I ask which store, because while I prefer CLI for updating and maintaining packages on my main device, the Discover store has always worked perfectly fine for me. And if you're using it on the Steam Deck level (SteamOS is immutable), then you don't really need to understand what a Flatpak is any more than you understand what an .exe is. Think if them as literally the same thing for your purposes.

And if anything, maintaining and updating programs is way easier than Windows as it does it all automatically.