Interesting to hear that plays through the Itch app count as downloads - I was wondering about those stats myself.
That said, it is also possible to right-click and select "Save Frame As..." which can be used to download browser games. I think it'll only really work with things like Twine games where the index.html file or whatever is all that's needed to play: if it's zipped up with other assets and whatnot then I'm pretty sure you'll end up missing those (though imagine there must be some way of getting hold of them since ultimately everything must be available to the player in order for them to run the game). I wouldn't be concerned about it, though: it shouldn't do any harm. The only reason I've ever downloaded stuff like that is to open it up and see how it's made.