GNOME doesn't really care about shortcuts on the actual desktop, but you could put those gamename.desktop shortcuts Steam created into ~/.local/share/applications instead, and then they will appear alongside your Applications in GNOME.
~ represents your home directory, basically an alias for /home/username/ so you wouldn't have to type that in a terminal all the time.
.local/ is a hidden folder (that's what the dot at the start does) and you might need to check if your file browser is showing hidden files and folders to see it.
To get more in-depth help in the future, you should probably state what Linux distro (Ubuntu, Fedora, Arch...) you are using too, different distros can have different ways of doing things!
Don’t know if it’s the answer you are looking for but if when you install the game you click add application menu shortcut it will add it to the menu that opens with windows key. I have never personally used desktop shortcuts on gnome since it isn’t included stock.
This, or the Tray Icons: Reloaded extension for GNOME, which adds the Steam icon to the tray bar. From there, you can click it and it shows a list of the installed games.
If you are new to Linux, GNOME has quite a different feel from your usual Windows flow. I personally love it and would never swap back.
(Off-topic, but If you would be interested in a gaming-oriented distro, I would recommend Nobara Linux - the official version looks fairly familiar for Windows users and it also comes with a bunch of gaming-related stuff preinstalled like Steam, wine dependencies, mesa drivers for AMD, etc)
Right click game in steam, add shortcut to desktop
double-click shortcut, game runs.
I checked the shortcut to see what the command is (right-click, properties):
steam steam://rungameid/648800
Is that what you have? The only thing I can think of is that steam isn't in your path, ie when you double click the shortcut it tries to run 'steam' but then can't find it.
If you open a terminal and type:
which steam
what's your output? You should have something like /usr/bin/steam