[native] Move Reanimated code from ChatThreadList to ChatThreadListSearch

Summary:
One more step in decoupling search from ChatThreadList.

Depends on D9206

Test Plan: Search experience continues to look and work as expected. Set logs in all expected callbacks to ensure that they fired as expected. Made sure that x-icon and "Cancel" button continue to look and behave as before.

Reviewers: ginsu, tomek, rohan

Reviewed By: tomek

Subscribers: ashoat

Differential Revision: https://phab.comm.dev/D9207