[lib][native] use processDBStoreOperations only from SQLiteAPI

Summary:
Cleaning this to have only one way of accessing this, I think it's beneficial to have a unified approach.

Depends on D13109

Test Plan: Play with app and make sure ops are processed and results are persisted.

Reviewers: tomek, inka

Reviewed By: tomek

Subscribers: ashoat

Differential Revision: https://phab.comm.dev/D13110