I am confused about your question. Maybe you are overthinking this.
Let's read your question again.
How is it that the itch app manages to run web games on linux
Will it help, if I mention that the Itch app is a browser? It's chromium based and has additional components for updating. And it can cache web games. But it is still web games. The operating system is not relevant. So what confuses me, is why you think there would be a wine needed.
Maybe I misread your question or there is information missing. You seem to try to do something and it is not working: nothing that i am trying seems to be working
What are you trying? Run the cached web game with a different browser, instead of the Itch app? That depends on the game. Not all will run this way.