[backport#16793]refactor: Avoid locking cs_main in ProcessNewBlockHeaders

Summary:
The locking was added in a previous pull request during the review phase, but the reason for this lock was later removed during the same review.

Backport of Core PR16793
Depends on D7634

Test Plan:

cmake .. -GNinja -DCMAKE_C_COMPILER=clang -DCMAKE_CXX_COMPILER=clang++ -DCMAKE_BUILD_TYPE=Debug
ninja all ckeck-all

Reviewers: O1 Bitcoin ABC, #bitcoin_abc, deadalnix

Reviewed By: O1 Bitcoin ABC, #bitcoin_abc, deadalnix

Subscribers: deadalnix

Differential Revision: https://reviews.bitcoinabc.org/D7637