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.
Most of the time, by a large margin, all of the passengers who show up for a flight get on it. That's the premise for overbooking, for any given flight, some number of passengers don't show up (including for reasons like missed connections, which the airline still have to fix).
Sometimes, though rarely in the grand scheme of things, too many passengers do show up, and bumping is necessary. In the vast majority of those cases, someone volunteers for compensation.
It's also important to keep in mind that the Dr Dao incident wasn't caused by overbooking, at least not as commonly understood. The airline urgently needed to reposition a crew to allow another flight to operate, and decided that this need trumped the passengers.