It’s certainly possible to have an HTML file for download at itch. Or a .zip if you have images or audio that go with it. So if you see a wrapped Twine game for sale on itch you could always ask the creator to add the base version as well. The trouble with that is browser security stuff: Chrome won’t load external images and audio and scripts if you just open an HTML file from your filesystem. So you might have to run a web server (though there are easy-to-use ones for most systems). But there’s nothing about itch that stops creators from giving the option.
Yeah, that’s unfortunate. And there don’t seem to be any easy cheap paywall providers out there: all of them require a (usually substantial) monthly fee. And most of them provide subscriptions, not actual sales. I did find an interesting statement on Gumroad’s things you can’t sell on Gumroad page: it says that selling access to a website counts as a service, not a product, and is much more open to fraudulent chargebacks. So I wonder if that has something to do with it. There’s also the common attitude that “people won’t pay for access to a website.”
Not quite: a game page on itch is a single unit. You can set “tiers”: if you paid nothing, you can only see the browser-playable demo, if you paid $5, you get the full downloadable version, if you paid $10 you get the making-of ebook and videos. But you can’t sell separate files individually on the same page. So you can’t offer, say, a game with an optional selection of paid mods or DLC ($10 for the base game, $2 for each mod you select. that sort of thing). That’s supposedly the most-requested feature on itch that they haven’t gotten around to implementing yet.