Hmm, I've never really interacted with the desktop app myself, so I can't say for certain. I just downloaded it to see if I can figure something out, but I get stuck in a perpetual captcha loop, so I got no way of actually checking what happens when you download CH via the app.
What I'd suggest you do is, first of all, get a better browser than Microsoft Edge. That one's just terrible. :P
Beyond that, though, the itch.io app should store the game files somewhere on your PC. If you can figure out where that is, you should get the same access to the game as you would if you were to download it from the game's page itself.
Alternatively, check out my Patreon. Not trying to sell you anything. The game is available on there for free as well, but instead of downloading it from itch.io (which may be what gives you trouble with Microsoft Edge) you can download it via Google Drive or one of the other file hosts I use.
If neither of those options work for you, let me know and we'll figure something out. :)