... is easy! You can just upload the same .zip file textadventures.co.uk uses (as of QuestJS 0.5, the HTML file is called index.html).

It does give a lot of options, which does make it more complicated, but the basic principle is simple.
https://the-pixie.itch.io/professor-kleinscope

You never set the game's status to published, so it's not available for anyone but you.

Oops... Try it now.

It works perfectly now!