Connect the GameDashboard hotkey to the GameDashboardContext

Bug: b:273641056
Change-Id: Ibc4e5c5d45a020a84dd9a4c127326dfdaf380479
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4563346


Reviewed-by: Ahmed Fakhry <afakhry@chromium.org>
Commit-Queue: Prameet Shah <phshah@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1149453}