Having a square wave magnetic field is like having a square wave voltage waveform on a capacitor. The capacitor would require current spikes on the square edges, and the coil would require driving high voltage, narrow pulses onto the coil at the rising and falling edges of the square wave (and a small amount of voltage afterwards to maintain the current). The magnetic field in a coil is always proportional to current, so you want a square wave current source.
The shorted turn approach would allow you to get a square current waveform, but not magnetic field. On short timescales the shorted turn will develop a current opposite to the driving current on the main coil, so as to cancel the magnetic field. That's why the current can be allowed to rise fast (the magnetic field can not, without a lot of energy input).
A different approach to getting very sharp edge magnetic fields is to set it up as a propagating wave rather than a near field. For example any TEM waveguide terminated on one end will look like a purely resistive load looking in from the other end, and so you can drive sharp edges into it. The H field inside the waveguide at a point will have a square waveform.