I don't see how any of this is a limitation of React Native and not a result of poor app architecture. If a list is too big to fit in RAM, but not too big for the device, page it from the local SQlite DB. All kinds of web and mobile apps asynchronously fetch remote data while keeping a responsive UI. There are also service workers and other ways to prioritize UI.

The vast majority of popular chat and email apps keep at least some of the data on the server by default -- Gmail, K9 mail, Outlook itself -- so I don't see where there is proof that users want something different.