[web] Keyboard support for typeahead [11/13] - Use callbacks to interact with overlay

Summary: Using callback functions to change chosen button, accept it or close overlay.

Test Plan:
Tested how it all fits together. Results on a video (currently without scrolling overlay content - added in next diff):

Reviewers: inka, kamil, ginsu, tomek

Reviewed By: tomek

Subscribers: ashoat, atul, tomek, ginsu, kamil, inka

Differential Revision: https://phab.comm.dev/D6051