if youre in some kind of support, helpdesk, customerservice, projectmanagement, development scenario and you have it set up correctly, emails are mostly notifications that something happened in the tracking system. customers/employees only EVER email the abstraction layer, not personnel directly. you can either reply to the email and it gets automatically sorted by the software, or you can interact directly in the software itself. so in helpdesk software like enchant, a ticket stays open or held until it is closed. closing the ticket removes the entire conversation from visibility. a reply restores its visibility. if i move a normal email to a folder, and a person sends a replies to it, the conversation becomes fragmented. team inboxes delineate conversations and let you move the entire conversation including future replies. they also force standard practice on a team, so each person doesnt project manage differently. this gives you more transparency, and lets you read each others conversations so everyone can stay on the same page regarding trends or events.
talking to the developers, thats not exactly how they intended the software to be used, and internally they close tickets aggressively and move longer term tasks to project management tracking in trello.
on the gmail front, their tabs or whatever are a lifesaver. i only check the primary and update tabs daily, the rest is pseudo-junkmail.
tldr: to answer your question simply. if i get an email asking for something or that i need to act on, and it didnt come FROM some type of tracking/pm/ticketing software, i forward it to the appropriate software so it is added to the todo list. if youre on a team using task software like https://glip.com/ for instance, you add the task directly in the software, you dont email the person.