With the onslaught of emails and many customers treating email like texting with rapid back and forth, I switched to using Boomerang's plugin for Gmail that "pauses" your inbox.
All incoming emails are batched in a hidden category, and at your requested times, these emails are then revealed in your inbox.
With this background automation/gating, you can work through your inbox, reaching zero without the distraction of sudden responses, bounces, etc. And keeping you focused on subsequent emails.
The queued emails are accessible, they are not hidden - so if you need to see if a response came back right away, you can look, but your inbox remains very clean and quiet.
Boomerang can be run on the Gmail Inbox and Front will function like this, but if you have multiple inboxes, then the cost becomes high for so many.