Sysadmin use of email is often necessarily more or less interrupt driven
One of the things that people commonly do with virtual screens is to put their email client off in a separate virtual screen so they can ignore it and avoid having it interrupt them. As I mentioned when I wrote up my virtual screen usage, I don't try to cordon off email this way. Fundamentally this is because as a sysadmin, I feel my use of email is necessarily interrupt driven.
Allowing email to interrupt me certainly can derail my chain of thought when I'm coding or working on a hard problem. But at the same time it's absolutely necessary, because that email may carry news of an emergency or a high priority issue that I need to handle more or less right away. I almost never have the option of ignoring even the possibility of such things, so almost all of the time I have to allow email to interrupt me. The best I can do is contrive low distraction email monitoring so that when I'm in a flow state it distracts me as little as possible.
So I can stop there, right? No, not so fast. What this really argues is that email is a bad way of receiving high priority information like alerts. Because it mixes high priority information with much less important messages, I have to allow even unimportant things to interrupt me at least a bit just so I can figure out whether or not I can ignore them. If alerts and priority items came in through another channel, I could readily ignore email during high focus times.
(There are always going to be days where all I do is fiddle around with stuff and swat things as they come up; on those days, I'd read and handle everything right away.)
Of course the problem is that there is no good other channel today, at least for us. Oh, with work you can build such a thing (possibly outsourcing parts of it to companies who specialize in it), but there's very little in the way of a canned out of the box solution. Plus there's the problem of getting people use your new 'urgent things' channel when they have an urgent thing and of course not using it when they don't have an urgent thing (with the associated issue of having people know whether or not their issue is urgent).
(Life is likely somewhat easier if you can assume that everyone has a smartphone, perhaps by issuing them one, but that is not something that's true in our environment.)