[web] clean up thread picker modal

Summary:
This diff cleans up the one off styles in the thread picker modal and replaces it with styles of the redesigned modal.

Linear task: https://linear.app/comm/issue/ENG-5943/extendmodify-the-modal-props-api-to-follow-new-modal-designs

Depends on D10320

Test Plan:
Please see the screenshots below

default:

hover:

Reviewers: atul, rohan, kamil

Reviewed By: rohan

Subscribers: ashoat, tomek

Differential Revision: https://phab.comm.dev/D10321