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How do you try it out in the web browser? There's no obvious link on the website

It is part of Simon Tatham's puzzle collection, it also has an app on F-Droid:

And the sgt-puzzles package on Debian/Ubuntu, possibly other distros package it too.

I haven't published the web version of the app yet, maybe I will in the future

Your description above says it's a supported platform FYI

Sorry, my bad