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Now what you should really do to make this more interesting that just a concept demo of what amounts to an immediately obvious gimmick, is to use the underlying hyperbolic geometry, but instead of making that the obvious focal point of the game/ui, make that geometry a hidden component of the game and force the user to deal with it indirectly or as hidden information.

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I have to point out that Hyperrogue is far from a tech demo. Though the controls are simple, there's some 60 lands each with fascinating rules and emergent complexities. https://www.roguetemple.com/z/hyper/gallery.php

Regarding making geometry a hidden component: it, kind of, is a hidden component in HyperRogue. I mean, yeah, it is in the name. But people play HyperRogue without understanding what hyperbolic geometry is. They think it is a game taking place on a sphere, or a normal Euclidean map with some kind of fish-eye projection. On the other hand, I do not think it is really possible to hide it from someone who knows how these things work, as the perspective works very different in hyperbolic geometry.

Which is actually quite cool: while so-called "non-Euclidean" games do all they can to show how weird they are, the actual non-Euclidean geometry pretends to be normal (but actually it is way more weird).

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