Once true email-to-chat interaction is a supported default based on simple start-chat metrics for email messages and a More useful "reachable by" chat interaction setting - #4 by monperrus is there, the hard necessity to show emails to catch contact requests should actually be softened a lot, because chat qualifying emails will then be simply received automatically.
Nevertheless, showing emails is still a very necessary feature to be able to check one’s email inbox, and so much required to be able to reply or answer to a classic email with a quick (interim) chat.