I don't think it's because you have a new iPhone. I have a new one 14 days ago and MC is always on, I wrote about it several posts before, look for it. What I have seen is that when you restore, MC just work for 10-15 seconds then it goes away, minutes later it comes again, and it is working like this for 24-30 hours and when you fully charge your iPhone from the wall (not usb to pc), MC comes on again to be there forever.
I have seen it because I have restored my iPhone 4-5 times, each restoration my iPhone is set like new iPhone and sync all my apps one by one, I never have used backup when I restore, so I'm very sure it doensn't care if you restore or not or if you use or not a bakcup to MC be always on.
There is a last chance to solve the issue: if your new iPhone is a really new iPhone (everybody knows that when Apple changes your iPhone they don't give you a full brand new but a refurbished one) it is possible that the full brand new has another component different to the previous iPhones. How can you see it? In the back of your iPhone, where are all the logos of regulations (FCC, !, CE), if you see a big "0682" it means your iPhone was made 1-2 months ago; if you see "0682" just before CE in a tiny font, it was made several months ago. So, if it is with the big "0682" may be it has another hardware and it's working correctly.
Please let me know what kind of "0682" do you have.