The iPhone belongs to ATT.
The fact that it has a fruit logo on it is irrelevent. You are paying to use ATT's system get used to it.
When apple does a deal with FOX for the next Mac you will have a machine that can only access Fox.news not the BBC.
But it will have a nice smooth case - so don't worry.
Why is Apple signing exclusivity deals at all in the US? Down here (Australia) they just go with everyone. The do make them bend and turn and invent new plans that include a calls/texts/data package that's more to Apple's liking, but they go through everyone.
I suspect they had to, initially, and the only terms they could get were for several years (basically until now).
Every US phone is aligned with a carrier. It's not immediately clear to me why that is, but there you go. When the contract is up, Apple has a serious opportunity to shake that notion, but I'm not sure if they will.