Just in case someone is looking to do this on their handheld, I figured out how to make web apps work nicely in game mode.
I use PWA (Progressive web apps) on all my systems, that’s Windows, Linux and FydeOS, and now I use them on Bazzite on my Legion Go.
Recommend you connect a Keyboard to your console to set this up, once done, you wont need it.
First step is to go into the desktop mode, create your web apps with Brave, don’t log in as you will need to do that in the game mode, personally, I create a new profile in Brave for my web apps, specially if you want to use any google services like YouTube, if you do this, then create your web app using that profile, and add any extensions to that profile, like SponserBlock ect…
Next add those web apps to steam, either by right clicking the web app and choose add to Steam, or by going to your Steam library and adding as a non Steam game.
Once that’s done, then go to the properties of the newly created Steam item, and add the following to the end of the launch options:
“–start-fullscreen”
So it should look something like this:
“app-id=wdsucsufbwfbwuewifqdsdcfv” “–start-fullscreen”
Once that’s done, you can return to game mode, then go to the properties of the app again, and select “Native” in the resolution options.
Once done, then launch the app, you will see its now full screen, if your on the Legion Go, the resolution will be way to high so you need to zoom the app, this is why you need the Keyboard connected, just hold down left CTRL and press + or - key until you are happy with the zoom amount, after this it will always start with this zoom level, YouTube needs to be around 150 to 200% to be usable as example.
Next, before logging into your web app, press the Steam button and go to the control configuration:
Click “Edit Layout”
Bottom left click “Action Sets”
Under Action Sets, click on “Default”
Select “Add Always-On command”
Then click “Add Command” (will be greyed out, but you can still click on it)
Then go to the “system” tab
Select “Touchscreen Native Support”
Job done.
Then leave this an go back to you app, and it should perform perfectly and operate like an Android app, only better 
Note: Brave and all Chrome based browser’s will not auto trigger the onscreen Keyboard under Linux, don’t ask me why, they just wont, under every single Linux version this is the case, only Firefox based browser do that, so you need to swipe up from the bottom of the screen to invoke the keyboard when needed.
Hopefully this helps.