In that case you do you both object to automated spam filtering etc. as well? Do you think automated processing of emails is OK or not.
Do you think it is not ethical for someone to get an assistant to open their letters and process them (decide which to send a form reply to, which to put in the bosses in-tray, etc.)? That is a MUCH greater invasion of privacy than Google's purely automated processing.
What about the other examples of automated processing above? They are just as great an invasion of privacy. Are the ethics of those questionable? Should Gmail also not have spam filters?