I think we used to pass the Profile* into the extensions layer via AutomationEventRouter. However, we don't do that any more.

AFAICT the only use of the BrowserContext is an early-exit here:
https://cs.chromium.org/chromium/src/chrome/browser/ui/aura/accessibility/automation_manager_aura.cc?q=automationmanageraura&sq=package:chromium&l=222

Eliminating this would simplify the code as well as make porting to mash easier.