I'm relatively new to using Itch.io for anything other than in-browser games. I decided to make an account and install the itch.io launcher, because I'd hoped that there would be something built in to run windows games on linux, like how steam has proton. I can't find anything in my settings (though that may be because I'm personally bad at finding things); and furthermore, itch.io games very specifically do not seem to be working with wine if I install them directly from the website.

Does anyone have any advice? I'm running ubuntu; everything is the latest version (I updated everything - kernel, OS, wine - as part of bug-testing in the past few hours; I only just installed the itch.io app in the past half an hour or so).

Proton is not built into Steam. It is extra software, that is a fork of wine. You could even compile it yourself, if you wanted to.

If you have Steam on your linux, you should be able to run non-Steam games with the bundled Proton.

There also seems to be ways to run Proton without having Steam installed. Using wine or other forks directly might have different results in terms of compability.

Regarding the Itch app, that is mostly a browser. It has a configuration to use a poor man's sandbox on Windows, and it caches browser games better. Otherwise there should be no difference between downloading and running games with the app or from the website, besides the launcher aspect.

So if the games you downloaded do not run with your wine, there is nothing Itch specific about it. They just do not run on that particular wine fork. Try another fork or a virtual machine. Pester the dev to release a linux build of the game. Play a different game.

You have to install DXVK to Wine (Vulkan support). Most games on itch uses Unity or Godot. Those games run fine under Wine. More games made with Unreal Engine run good too. 

For me the biggest problem only occurs when trying to run games that use .NET, but this can also be solved by installing the appropriate libraries in Wine.