Headphones, situational awareness, and even diversity
It's a given that headphones are a good way to block out the world and focus on things for a while. That's fine for times when it is appropriate. I'm more concerned about those other times which may be overlooked.
Headphones utterly fail in a support team situation in an open setting.
It's pretty common for such teams to have tiered support, such that your "level 1" techs take most phone calls and try to triage things, your "level 2" techs deal with walkthroughs and most ordinary issues, while "level 3" handles everything else by definition.
Let's say a customer calls in with a question about their setup, and the level 1 tech can't answer it. However, the level 3 tech sitting right next to them overhears the situation and jumps on to look at the machine. It's totally possible for that level 3 person to get the answer and then Jabber it back at the other tech. That person then relays the info to the customer, and that's it. It's done.
In the best case scenario, a bunch of good things happen here. First, the customer gets their answer from the same person who picked up the phone. This usually blows their minds, since many of them are used to being transferred and being put on hold and bouncing around through a bunch of menus before they get anywhere.
Next, this is far faster than the case where the first tech has to go find a second tech and then transfer the call. Every minute you burn doing that kind of meta junk is another minute which is not available to help people with real support issues. Eliminating that time is good!
Finally, there is a non-zero chance that the level 1 tech on the phone will learn from this experience. On that first call, they may not know how to check if a given service is running on that kind of server. Afterward, they might learn from whatever the other tech ran. That means they might just be able to do it themselves in the future!
After this happens a bunch of times, odds are, that level 1 will not be a level 1 any more. They will progress up to level 2 and will start taking on bigger and better things. They also probably won't have to answer nearly as many "raw" phone calls, since their new position has that coveted level 1 firewall in front.
This is what you want. You want people to learn, and grow, and move up.
If your senior people who know all the answers won't (or can't) pay attention to what's going on around them, it might just affect the whole team. Incidentally, this makes an excellent case for hiring people who are both technically adept and are able to pick up on things like this. In other words, those folks who just zone in on the screen and lose track of everything else around them probably won't be too helpful here.
Put another way, this is why diversity on teams is good. Without it, you might not even realize what you're missing. It may not even be apparent that there is another way to handle a given situation.