[services] Backup - Pull more mutual code from base reactors - Apply in client reactors

Summary:
Depends on D3785

https://linear.app/comm/issue/ENG-916/consider-pulling-more-mutual-code-from-the-async-reactors

Applying inheriting the BaseRactor in the client reactors

Test Plan:

cd services
yarn run-backup-service

Reviewers: palys-swm, ashoat

Reviewed By: palys-swm, ashoat

Subscribers: ashoat, palys-swm, Adrian, atul, benschac, yayabosh

Differential Revision: https://phabricator.ashoat.com/D3786