itch.io

True.  I also checked the code today and saw that it is supposed to allow windows games to be added directly. I am on v26.22.0, so it must be a bug of some sort.
UPD: Just checked on Debian (though i used WSL so anything could go wrong) and it also doesn't show the switch:

Probably not the solution you are looking for but I recommend just downloading the regular non-Flathub version. Since flathub apps use sandboxes you'll either have to do some permission voodoo (flatseal or smth, i think its easily googlable) or manually add games to Steam.
I am also on steamdeck and that's the solution I opted for.

Having the same issue. Interestingly enough, when I run itch through Proton 4.11-13 it for some reason does show those options.
Running Itch directly: 

Through Proton:

However I can't apply those changes since it requires exiting Steam which in turn exits Itch before it can do any changes.
I assume Itch app allows adding games directly only if it thinks it is fully compatible. And since those games are built for Windows, they can't run directly, therefore it does not show the "direct/via itch" switch. Maybe Proton makes it think it is on Windows and so it thinks it can run natively on the machine so it starts showing this option. Anyway its very confusing and probably should be changed to be less constrained. I wish the app would allow you to do more and just give you a banner like "it might not work" instead of fully disabling useful features.

Well, the game is built for Windows, so it checks out. Add Wine\Proton layers and SteamInput's magic on top and the behaviour will surely be hard to predict.

I wanted to try playing this game on my Steamdeck, and through trial and error i found that Proton 4.11-13 is the latest version that runs it. I hope this helps someone. Also, I created this Steam Input Layout (steam://controllerconfig/3593839953/3818172523) with basic mappings. (Nvm, I didn't see it has gamepad support, though its finicky. You'll firstly need to use some layout to enter settings and then choose the gamepad, which for me was the 4th one, and only then you can switch to regular gamepad layout)