Move common logging start code to init/common

Summary:
This is a partial backport of core#21732
https://github.com/bitcoin/bitcoin/pull/21732/commits/5bed2ab42c4f1a820468f7005ce62e39001f6611

Depends on D12562

Test Plan: ninja all check-all

Reviewers: #bitcoin_abc, Fabien

Reviewed By: #bitcoin_abc, Fabien

Differential Revision: https://reviews.bitcoinabc.org/D12563