I was concerned your stepping motor would not have enough drive power. However, there is a gain since you will be using a screw...
I often use surplus windshield motors for high torque applications (easy to get locally).
because I need some idea of how far I've spun the corkscrew
You could have a limit switch to detect the stop/rev/crash position
unnecessarily complex
I don't think so.
Note: you can use PWM for speed control on the gear motor.
If you are just starting to use motor drive circuits the gear motor is a good choice.
A DC motor is usually smaller when compared to a stepping motor considering drive power.
They also use a fraction of drive current.
Using a stepping motor as a first project is OK if you insist on using one, rather if you want to learn about them.
The whole reason I do things like this is to learn.
If we aren't learning, are we dead? To learn is to be young.