[Switch Access] Separate init() and start()

This change is a pure refactor.

AX-Relnotes: n/a.
Bug: b:261227533
Test: cq
Change-Id: If2579970c933511541c6f5b1c0cf3ebefea0dd07
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4286141


Auto-Submit: Anastasia Helfinstein <anastasi@google.com>
Reviewed-by: Kyungjun Lee <kyungjunlee@google.com>
Commit-Queue: Anastasia Helfinstein <anastasi@google.com>
Cr-Commit-Position: refs/heads/main@{#1184821}