Although I haven't even started training yet, I feel obliged to contribute my thoughts. I'm of the opinion that a simple medical should be compulsory for all who intend to fly. There are too many "silent" conditions that, as the name implies, go unnoticed and could have detrimental effects. A simple example would be uncontrolled hypertension (high blood pressure). This is a classic silent disease which predisposes you to strokes, TIA's, myocardial infarctions (heart attacks), angina, acute bouts of shortness of breath, peripheral vascular disease (cramping of your legs as you're trying to coordinate a turn on final...). These potential complications all occur suddenly and would immediately hamper your ability to pilot the aircraft.
A simple medical would pick this up and the recommended management would be to achieve adequate control for say, 3months, and you would be allowed to fly.
Granted, these risks apply to road users as well, but it would simply be too much to implement. Piloting an aircraft carries significantly more risk, in my opinion, than driving a car, though I accept that this is arguable.
Again, a simple medical would not encompass as much as the Class 1, but it's a good screening tool.