So I first made the app web-only, which is what I know best. Used CSS animations and they were smooth!
Then converted the project to React Native (because I wanted to learn how to make a RN app). But because I wanted to keep using the web app I used react-native-web, which had also been intriguing me for some time.
Result? Animations became much slower! react-native-web basically polyfills RN's Animated library [0], so maybe it's not very optimal. But to be honest, even when running the native app the animations seem janky (tried on Android via Expo and on iOS simulator).
So react-native-web is a great project, but I'll likely not use it again for a similar project. I'm a web believer anyway, so I'll probably do just web apps going forward unless some use case really benefits from native APIs.
[0] I never looked at the source but I assume the code for web animations is here https://github.com/necolas/react-native-web/tree/3fa18becc76...