[composebox] Clean up composebox input fields.

There were duplicate fields in the composebox that were not needed.
Instead of updating this.$.input.value, we should use this.input_ should
be bound to the value of the input. This way we only need to keep track
of one value.

Change-Id: Iea170c5acddeb8d9a4aa5c98a7300a99a5a3877b
Bug: 450324345
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/7022493
Commit-Queue: Nihar Majmudar <niharm@google.com>
Reviewed-by: Marlon Facey <mfacey@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1529034}