A lot to address here:
You were off for a bit before starting again, this program seems a bit aggressive - too much running at a 5k pace without any slower miles. While off days are good, easy days are good as well, active recovery is important as well, it helps acclimate your body to training. I agree that the program is a bit aggressive, but my biggest problem is how incomplete it is - pretty much you're either running 5k pace or you're taking days off.
This program will not get you in shape for racing, and shouldn't be designed to, it should be designed to give you some base fitness coming back from injury before starting a real training program. There is no proven "only" way to train. A good training program utilizes different kinds of runs for different parts of your fitness - medium paced runs (I'd call these tempo runs) are only one tool, you need long slow distance, shorter easy recovery days, intervals, and sprint work, all at different times within a training cycle.
You don't need to be making these arguments to him, your parents should - you're just a kid, it isn't your place. His temper shouldn't matter - YOUR PARENTS pay the bills, he is working for YOU, if there are concerns, he should address them in a professional manner. You might consider a different PT if you still need one.
Supporting the team is great, but ultimately doesn't matter. What matters is having your best race. If you do your best, that is the most you can do for the team. Running isn't a team sport (unless you happen to have a baton in your hand).
You can change whatever you want. This guy is just a doctor, he's not the police, if you are healed, train in an appropriate manner - if your father and you both think this guy doesn't know what he's doing, do what you think is correct.