[keyserver] Handle new session state sync

Summary:
Iterate over specs and fetch appropriate data.

https://linear.app/comm/issue/ENG-4632/migrate-state-sync-to-a-spec

Depends on D9049

Test Plan: Bring back the old code and compare if the same payload was generated after closing the app, deleting a session and opening the app.

Reviewers: kamil, bartek, inka

Reviewed By: kamil

Subscribers: ashoat

Differential Revision: https://phab.comm.dev/D9081