There is an unofficial Rust binding for SFML:
https://github.com/jeremyletang/rust-sfmlKeep in mind that you need actually 3 libraries for SFML if you use Rust: Rust (API), C (binding) and C++ (implementation).
How well Rust is supported on Android/iOS is another story, and you would need to clarify that yourself.
Would be definitely interesting if you went this route and could share some experiences here!
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Keep in mind that going with C++ is probably the easier route, but you can also do that as a first step and in a second step integrate Rust.
Is it possible to have a native Android/iOS app that has a central widget, and the core lib to render on it?
If I understand you correctly, you would just want to render with SFML and do the logic from your library. I don't see why this shouldn't work.
For tutorials on how to use SFML in iOS and Android, have a look at the wiki:
https://github.com/SFML/SFML/wiki/TutorialsBoth iOS and Android ports are not that actively maintained, so it's possible that you need to search a bit if you hit any roadblocks.