Multi speed motors and gear boxes

Now the question
What if you want to accelerate well and go fast?

lets all agree it takes the same HP to go say 100MPH regardless if there is a gearbox, multispeed motor or not

So there is no free ride here if the motor can't put out the power no gearbox or multispeed motor is going to get the bike to speed.
The power plant needs to be in a proper operating RPM to be efficient and output the power needed for the speed required.

If you have a wide torque curve as most electric motors do then maybe you don't need a gearbox or multispeed motor but even an electric motor has a most efficient RPM and that is up around 60 to 80% of the max RPM (BEMF) So even thought a gearbox may not be necessary, it will result in more HP because the motor is operating in its most efficient region. For a multispeed motor implemented correctly this is also true. Since a high wind count motor has more amp-turns it has higher torque for the same current (which will result in overheating if sustained) or half the current for the same torque (the correct way to look at it) but it trades speed for torque and will be voltage starved at half the RPM, So the 60% of max RPM is reached early and so does Max efficiency. Then if one switches to half the turns and doubles the KV of the motor it is now possible to get higher RPM and again achieve max efficiency.

In Conclusion more power is achieved with both the gearbox and the multispeed motor because the motor is spending more time at higher efficiency really no other reason, but the gearbox has losses that make the gain in motor efficiency sustainably less power to the tire. The gearbox losses are not present in the multispeed motor design.