Using DFT to determine control system bandwidth from user input
Your basic approach seems correct! However, step 2 is not a good idea since taking the square of the signal is a non-linear operation. Taking the square of the FFT to determine the signal power is correct though.

For your profs. question it depends on what you have measured and at which sample rate. I assume it is the steering wheel angle and therefore it should be possible to answer it, but there will be some pitfalls as most likely the steering movement is not in a sinusoidal form. However, the basic frequencies in the movement will be related to this.

Given that you redo the estimation without the squaring of the signal, the ten times safety margin should be more than enough.

To further convince you and your prof. I would suggest bandlimiting the measured data to your estimate by using a very sharp FIR filter and compare visually as well as downsampling the data and compare.