This follow-up commit replaces the current left sidebar channel list
filter input implementation with the redesigned input component.
Note: This commit removes `clear_search` method from `stream_list.ts`,
since the `.input-button` onclick handler over at `inputs.ts` handles
the clearing of the filter input fields by sending an empty text input
event. This input event triggers the filter update handler of the
module, in this case the `update_streams_for_search` function and
resets the filter list, eliminating the need for a separate per-module
input clearing/resetting workflow.
Fixes part of #34476.
