Migrate to React Native New Arch

[retrixe]

Known issues:

• [ ] Modals cut off near the bottom. React DevTools can't tell that it's cut off either, for some reason. https://github.com/facebook/react-native/issues/35804
• [x] Tab labels aren't fully upper-case. https://github.com/facebook/react-native/issues/39524
• [x] The inverted chat messages view isn't functional. ~~Our existing workarounds for this most likely don't work anymore.~~ React Native 0.72.4 removed the need for a workaround in both old and new architecture.

Opens possibility for TurboModules-powered native connection back-end, preferably in Rust:

• https://stackoverflow.com/questions/74977723/write-react-native-turbo-module-in-rust
• https://reactnative.dev/docs/next/the-new-architecture/cxx-cxxturbomodules