The Itch app from the store won't let me install Windows games, only Linux titles, and I'm having the same problem with the client downloaded from the website directly. Since most games on Itch aren't designed for Linux, I really can't do much. Did I miss something?
Viewing post in How to play itch.io games on the Steam Deck comments
Take a look at this thread here: https://itch.io/t/694459/please-allow-installing-windows-versions-under-linux
Specifically, this reply from Fahrstuhl:
Put your Steam Deck into Desktop mode by pressing the Steam button, clicking Power and select Switch to Desktop. Open Firefox, download itch for Linux from https://itch.io/app and run the installer (double clickitch-setupin theDownloadsdirectory). Then right click on the itch icon that was created on your desktop and open with Kate, a text editor. There you replace the line that says:
Exec=/home/deck/.itch/itch %U
with a line that says:
Exec=env PATH="/home/deck/.local/share/Steam/steamapps/common/Proton - Experimental/files/bin/":$PATH /home/deck/.itch/itch %U
On the next start of the itch client in desktop mode, you can download and install Windows games.
That allowed me to install Windows games through the app.