Earlier quoted context omitted.
You can but they'll likely only starting preparing the Rezept when you arrive, and you'll still need to wait 30 minutes. At least that's how my Hausarzt works
You should change your Hausarzt. They can be so careless only because enough people tolerate such behaviour. I just send an email what I need, they reply to me the same day or tomorrow that it's ready to be picked up. I got there, and get it in 2 min.
In most places it is hard enough to even get an Hausarzt to being with. You might just be lucky to live in a bigger city where you have the ability to choose.