[web] Use message search context in message search modal

Summary:
issues: https://linear.app/comm/issue/ENG-3466/add-some-state-for-keeping-the-search-query, https://linear.app/comm/issue/ENG-4109/add-search-button-to-search-modal

The query is saved only once the user has pressed "Search".

Test Plan:

(for the purpose of recording this I bound the pinned messages bar to message search modal)

Reviewers: kamil, kuba, michal

Reviewed By: michal

Subscribers: ashoat, tomek

Differential Revision: https://phab.comm.dev/D8274