To me that sound like there is something wrong with the install. There is no reason why the radio shouldn't come on when you start the car. Even if you remote start the car or turn the key it's all the same the car is turning on. When I remote start my car everything turns on fine and thats the way it should be. From the sound of things one of a couple things have gone wrong here.
Either some of the wires have come loose and aren't making a connection all the time. I really doubt this because then it wouldn't work at all sometimes. But it sounds like it starts everytime for you.
The Remote start brain has going bad and isn't working right. Again I doubt this for the same reasons above.
The connections from your radio to the car are loose. I would bet on this, I would double check all connections beween the radio and the car. My bet it's a grounding problem or something is just wrong with your radio.
Look at it like this. The radio must turn on when and if the car is started. It doesn't matter if you remote start it or key start it. The only way that doesn't happen is if you turn the radio off before you get out of the car or the connections are bad to the radio. I'll bet you anything that if you test the wires be hide the radio when you remote start they will be the same if you key start the car.
What I say is check the radio there is a better chance thats the problem and the remote start is fine.