We can require more training for GA, but the training burden is already pretty high, and the GA industry is still shrinking last time I checked. Boosting safety requirements in the form of additional training for GA may save a few lives, but also those gains could be simply because there will be fewer hobby pilots.
The fact is that GA planes falling out of the sky just doesn't kill very many people. It obviously feels terrible to the families involved in a crash, but relative to automotive deaths, lung cancer, murders, etc, it is a tiny tiny blip in the numbers. Obviously being a GA pilot boosts your chance of dying in a small airplane crash, in the same way that learning to ride a motorcycle boosts your chance of dying in a motorcycle crash. The alternative of killing off recreational flying, is worse.