I think I discovered one workaround for when the rolling method is chosen as the type of start.
So, I'll describe what I happened.
The AI cars started moving slowly and with the hope the safety car went out of the pits or started moving from the start line (the initial position of the pace car depends on the track), but it was stuck, so the race wouldn't start and the AI opponent would go like 16km/h forever.
Then I hit the request pit key and what happened is that me and the AI opponent jumped to before the starting line already at a certain speed, then the game passed the control of my vehicle to me and after that, when I was near the starting line, the green lights went on and I managed to do an actual race. (in short, when I hit the request pit button, what happened was a fast rolling start)
After that, I did another test. I set up a race in practice mode with 5 minute warmup. At the warmup phase, the AI car wouldn't move. But then I tried the following:
After the pit's traffic light is green during the warmup, I returned to the race monitor. Then I hit the race now button and immediately after that, the request pit button, and the AI car (opponent, not the pace car) started moving, did a couple of laps before stopping at the pits for some moment, and off he went to the track again.
So, what I can conclude from this is that the problem I described in the main post is not entirely due to the AI implementation, but also due to some code the developers forgot to delete when they we're implementing and debugging something about the AI implementation, or something like that, which make the executable wait for some user input instead of doing it automatically.
I hope this helps to solve the problem