Uhhh... how do I say this... Basically I missed a button when uploading the game and instead of "Downloadable" I should have selected "Play from browser".
What you can do: Technically you can upload the game in itch.io as your own project (maybe don't publish it to the public, but keep it as a draft) and you can play it in browser. Here is a tutorial I found that should explain each step https://www.construct.net/en/tutorials/publish-c2c3-html5-game-itch-2363/page-3. Really sorry for the mess... also will update the page now with explanation of what works in the game at the moment, so people are not completely lost.