If markets can be stabilized by algorithmic trading, why aren't markets more stable today than they have been historically?

Central banks, through the manipulation of interest rates, cause the boom bust cycle. Every bust in your chart took place during the era of the Federal Reserve. By manipulating interest rates below the natural rate of equilibrium, the Fed distorts the true cost of credit, making unprofitable ventures appear profitable. Each bust is the market's response to clear out the malinvestment that was caused by monetary manipulation.

Eliminate central banks and let the market regulate itself. You won't have a boom/bust cycle, and markets will appear more efficient.