Never seems to have much benefit for me, because I'm typically the one who answers the phone/email/etc, and "the datacenter team" never does. No one is happy with that, and I still get the calls/email/messages, demanding the 'fix' or ETA. I would rather things be more under my control, because I inevitably am still 'blamed' when things go wrong.
And receiving a call and telling: "here's datacenter emergency phone, please ping them" or forwarding mail to datacenter team with "can you give ETA to the business" is an order of magnitude easier than trying to concentrate and resolve the actual problem while being interrupted every 5 minutes and not having required level of access to actually do it, even if the knowledge exists.
Though in most of the companies, business will not call you, but some poor business analyst or project/product manager, which will learn quickly to keep you out of the loop and don't waste anyone's time if you are consistent with answering "have no idea, ask datacenter guys".
You seem to have exceptional customers. Like the parent comment, my customers wouldn't have that. I'm their point of contact, and they want me to do the legwork and provide them with updates. They want to complain to me directly, often citing that they have enough problems with their business to care about problems my business is facing. We'd be out of business if we insisted they talk to our vendors directly. They would straight up laugh in our faces as they fired us, and frankly, I'm with them, since I wouldn't put up with that from our vendors either.
In rare cases that some C-level break through and complain directly to me, it's usually easy to explain that if he wants a solution or an estimate fast, he should be dealing with datacenter team, since I don't have any leverage upon them, and he can complain directly to the IT-Infrastructure Director, to whom I don't even have direct contact.
Honestly that's the best thing you can do for on-the-job sanity. I understand that people feel they have a right to abuse others, but at some point you really don't have to take it.