Based on personal experience, I doubt this is true. Ask yourself how many times have you done this? I've flown hundreds of times, but have only missed a handful of flights. So the collision rate should be very close to the actual overbooking rate.
I haven't been able to find any figures on the actual rate of overbooking. But I did see a statement that 11% of bumps are involuntary. And the overall rate for involuntary denial is 0.09% according to this article, for instance: https://www.ft.com/content/e4cb5744-1e9d-11e7-a454-ab0442897....
That suggests that the overall rate of overbooking collisions is around 0.9%, which is close to my overbooking estimate of 1%. Even if I'm off by an order of magnitude, the impact on fares would be no more than 10%.
Eliminating the fraud exemption for airlines wouldn't necessarily mean removing seats; as I pointed out, the seats could be filled with discounted standbys.
Lots of other businesses sell seats without overbooking, I hardly think it's an unsolvable problem. Airlines got a fraud exemption because they could, not because it was the only solution to the problem.