[Due for payment 2025-09-16] Follow up: make SEARCH_REPORT route compliant with routing philosophies

[roryabraham]

Coming from https://github.com/Expensify/App/pull/68536/files#r2301355739

Let's refactor the SEARCH_REPORT route to be compliant with routing philosophies.

[VickyStash]

Updates:

• Prepared Draft PR
• Next step is preparing the recordings

[VickyStash]

I've opened the PR for the review cc @hungvu193

[roryabraham]

Approved the PR, awaiting C+ review from @hungvu193