You shouldn’t want to, this plastic gear serves as a kind of mechanical fuse to protect the motor from failure. If the system is overloaded (or worse freezes to a stop) the gear will shear off protecting the motor from overload. This part was not designed as “planned obsolescence” as the rumor states but was actually a safety feature as an easily accessible and cheap plastic gear is a much better replacement than a motor.
I think it is driven off of the 12V line with a resistor. You should be just fine to remove it.