[RN] Update library's RN dependencies to latest

[c0mtru1se]

Task specification

With the introduction of stable New Architecture in React Native 0.76, we should focus on developing the lib against the latest React Native core.

This is most important in regards to Codegen, as libraries depending on older RN might not compile properly. Codegen is now finally considered stable (RN#46208).

We need to update our library's dependency on React Native to 0.76+ and test against this version.

This ticket should logically serve as a prerequisite to [#217].