I wouldn't say it'd be unsurprising to build a turing complete game - building a game (or like, any software) is an exercise in reducing the functionality of a computer, if a machine can originally do X things then your program can do Y things where Y ⊆ X - I think that programs maintaining turing completeness sort of need to go out of their way to be so feature full.

For instance, HTML[1] is (IIRC) not turing complete, the computation power once you step into a layer of pure HTML prevents you from ever assembling a turing machine - no matter how many gigs of HTML you can pump out you'd never be able to produce a turing machine[2].

[1] Pure HTML, HTML + CSS apparently is turing complete.

[2] This disregards merely using HTML as a data definition format and using other logical components to enable the construction of a turing machine - a turing machine's tape is as simple as can be, so we don't really care about storage formats that can replicate the tape portion, we care about things that can replicate the full machine.

Yes, it makes sense that the program can do a subset of what the computer is capable of.

Regarding HTML, I wouldn't say it's analogous to a simulation game. Maybe the map, but the running game is more like HTML + CSS + JS

If you take bugs and security issues into account then almost any layer built on top of Turing complete software is Turing complete