Adventuron games hosted on itch.io have a similar problem, except that (in my experience) they just display a blank screen with no error message. (For instance, The Familiar.)
I have a fiddly workaround for both Twine and Adventuron cases, that doesn’t involve allowing third-party cookies:
(I don’t know how stable these URLs are – probably they can’t be guaranteed not to change – but that one has been stable for a couple of months, at least.)
You can also get the url by pressing “Run game”, right-clicking on the frame, then selecting “This frame” → “Show only this frame”. (This doesn’t seem to work with the Adventuron game you linked, but it does with Twine.)
As you’ve seen yourself the domain was stable until it was changed very recently. It should be stable for a while now. The other moving part is the /7625275/, which is updated on each release.