A solenoid isn't going to have enough control granularity to tune a Herbie overflow. I find that I have to get my gate within a pretty small window for my Herbie overflow to be silent.
As you said a solenoid is opened or closed, which doesn't give me much granularity. That leaves me a few options.
1) Open and close the solenoid often. That's going to be noisy and I doubt it would work very well. I'm not even sure solenoids are meant to be used like that.
2) Add some complexity to the electronics so that the solenoid can be partial opened. This seems a lot more difficult to me than turning a gate valve with a stepper motor.
Your method would most likely be cheaper but you're relying on something mechanical which can be fairly harder to control. How much do you adjust the valve to get the flow you need?
I was thinking of using a PID control loop for controlling the stepper motor. The PID loop would optimize for a specific height of water in the overflow. Considering that the Herbie operates silently as long as the water is within 1" range means that the PID loop doesn't even need to be tuned very well since it doesn't have to be super accurate.
And what happened if the motor f's up and continues turning without stop and breaks the valve?
I can set a maximum current on the stepper motor driver that will prevent the motor from having enough torque to break the valve if it turns beyond the range. In the case that the valve is turned all the way, the motor will start torquing up and start consuming drastically more current. This drastic increase in current will blow a resettable fuse causing the controller to be turn off.