Yes, a missing OVP will definitely cause starting problems!
When I had mine out recently for a few days (to disable ABS - found out after that its not good for the car to run without OVP), the car was very difficult to start when cold, wouldn't idle etc. etc. Once it was warm it seemed fine. I don't remember it being a problem on a hot start, but I don't think I ever tried it.
Does your car have ABS? An ABS light on should have given away that there was some problem.
2.5-16v 1990 90,000m Astral Silver
2.0E 8v 1986 107,000m Black