@rob I’ve been giving this some thought lately. I’ve been doing both of these things for a while and I think a hybrid approach might work better.
IE - have a generic notification email either at your own email host or through Gmail, ie wpadmin_company (@) gmail.com. Now add a unique alias for each site you’re going to manage.
Why? Well I’ve seen some sites that get yanked away from clients only to keep sending zombie notifications for sometimes years to an admin email address that was left on the site. Even after trying to notify the former client no change. With this setup all you have to do is remove the alias from your main email address and the old notifications go away.
That’s my current thinking anyway…