But if you are a browser extension I would have expected you wouldn't need your own permissions at all: you would either be scripting the UI of the client to accomplish your goals or stealing its authentication.
If an extension only does actions involving the UI and data visible on screen within the Gmail webpage at a regular user pace, then I wouldn't expect it to strictly need the official Gmail API much. But this would mean the extension can't operate on emails that aren't on the current visible page, etc.