Most electric motors are most efficient and can generate the most power within a certain RPM range. (It's a lot wider than gas engines, but beyond a certain speed the torque falls off.) If you can drive at high speeds at a more efficient motor RPM, then that's less excess heat for the motor and inverter to deal with, and less current draw from the batteries.

For single-speed EV’s, the motors are operating very far away from their maximum efficiency point at high speeds. By upshifting, you can reduce the number of revs required at top speed, which should improve efficiency and reduce heating of the motors.

Alone it's probably not enough. In combination with other design decisions, it could be the key though. Take for example the smaller battery. Normally that would be a big limiting factor as battery size is somewhat indicative of maximum power output. But by shifting gears once through the acceleration curve, they can compensate for the lower max power. Furthermore, they could keep higher speeds with less constant power output.

The reason this matters is because a smaller battery will produce less heat overall, which will be easier to cool, and so on.

Anyway it's just a theory. The facts remain: the Taycan can do something the Teslas can't and this gearbox is the biggest single meaningful difference in design of the powertrain. It stands to reason they are related facts to some degree or another.

It depends what speeds make the motor run sub-optimally for a particular vehicle. The range 0-80 MPH is sufficient for almost all normal driving. If you want to make an efficient track car that can operate at high speeds optimally by all means introduce a gearbox. Its going to add more weight, complexity and maintenance though. I don’t think thats the target for any of Tesla’s cars besides the 2020 roadster though.