I do keep the same GP and GI docs.
I'm not allowed to have the relationship I describe. Even when I really do know the right answer, FDA laws require me to go to the altar of the MD to get a signed prescription. I'm forced to wait two weeks to get that appointment, even though I know the right answer.
(In reality, the doc is generally willing to call in the script on my say-so over the phone, but he's not supposed to do this.)