[web] Move safari db key init to database module

Summary:
Part of ENG-4844
Depends on D9105

Moved safari db initialization logic inside database-module-provider as it's more of an implementation details and simplifies future code.

Test Plan:
Test in safari:

• check if there's "Database initialization success" message
• check if draft are still there after reload

Reviewers: kamil, atul, inka, tomek

Reviewed By: kamil

Subscribers: ashoat

Differential Revision: https://phab.comm.dev/D9107