If it's possible to remove that envelope icon, then that would mean that people can't email other people but only send private messages through the speech bubble...
So were you to remove that envelope icon, then the logical result is that you won't get emails from anyone (and probably not send them either) but as it's not practical at the moment to remove that icon, the easiest thing to do is to remove your email from your profile or to put a dummy email address instead.
I can't remember if it's necessary to have your email when you register but if it is, once you register, you can easily remove/change it after. Admin have rightly mentioned that if you're not a member, none of this information is shown - i.e. you don't get the envelope icon showing up. Once you're signed in, your contact email, your website if you have registered the details etc. all display as icons...
And as far as I remember, this function has always been there.
Maz