As much as you wish there wasn't an emergency every time they are supposed to come your vet probably wishes it even more! They are possibly trying to balance family, their own pets and a practice that covers emergencies. Its a balancing act and I say that as someone whose christmas/birthday/weekend/weeknight memories are filled by emergencies pulling a parent away. I say this not to be rude but because I would guess that your vet has been in the same boat. My vet parent went 24 hours without sleep and did not see his family awake for days at a time. I am sorry but if the call is "he is down and rolling" or includes the phrase "prolapsed uterus" and the choice is attending that call or the gelding, the prolapse is going to win because the animal could/probably will die. They may cancel on you because most people don't want to vaccinate or have their horse gelded at 2 am. I know its not great but its life. I know that sometimes there are vets with less than a magical touch. Which is unfortunate, you can try requesting a different vet in the practice. If you have a training strategy you endorse share that with your vet. If you horse is bad on cross ties tell the vet so they can help you work in that system.
As for when they run late, its a pain I know. What you can do is call the office the morning or day of your appointment and ask how the doctor is running. You can also request that the doctor give you a call if they are running late. This should be a courtesy the doctor extends; however, sometimes its not. I try to remember that the doctor is balancing a large schedule that changes and is trying to treat everyone the same with respect to time and quality of care.
I am sorry if I come off as rude, I just have spent my life living that other side of the vets coin. I come at it with a different idea and I don't mean to offend anyone.