... it was also my understanding that most rockets are run fuel-rich to avoid an oxidizing environment that could erode the combustion chamber and nozzle
Reducing corrosion might be part of the reason, but I've always understood the reason for operating fuel-rich is to lower the molecular weight of the exhaust gas. Oxidizing the fuel creates heavy molecules -- CO
2 is heavier than CO, and H
2O is heavier than H
2 or H -- which results in a lower exhaust gas velocity. The downside of running fuel-rich is that the temperature is less, which also reduces exhaust velocity. There's a happy medium where we supply adequate oxygen to burn enough of the fuel to get a high temperature, but not too much oxygen that we drive the molecular weight too high. The optimum mixture that achieves the highest exhaust velocity is on the fuel-rich side of a stoichiometric ratio.
By the way, the Russians operate many of their engines oxidizer-rich in the preburner and turbines. I haven't studied their designs enough to know how they've overcome the corrosion issue, but they've apparently figured it out. I think their N
2O
4/UDMH staged-combustion engines operate oxidizer-rich, though the main combustion chamber is still burning a fuel-rich mixture. They route the oxidizer along with a small amount of fuel to the preburner, and then the oxygen rich gas is routed to the turbines. The turbine exhaust then goes to the combustion chamber where it is combined with the remaining fuel and burned.