[net processing] Move ProcessHeadersMessage to PeerManager

Summary:
Partial backport (4/9) of core PR19791:
https://github.com/bitcoin/bitcoin/pull/19791/commits/d7778351bf60a21925a97b7fc4e9df5541b6d995

Depends on D9037.

Test Plan:

ninja all check-all

Reviewers: #bitcoin_abc, PiRK

Reviewed By: #bitcoin_abc, PiRK

Differential Revision: https://reviews.bitcoinabc.org/D9038