A bypass cap can be helpful if you have high frequency noise, as capacitors have some internal inductance that can make them ineffective at high frequencies. The Quickie power supply operates at 0 Hz, so this isn't a factor.

Batteries have very, very low internal impedance. I would expect the string of Alkaline 9V batteries in the Quickie to be under 10 Ohms at DC. As the frequency across the power supply increases, this resistance will go down.

A 0.47uF capacitor presents 340 Ohms of impedance at 1kHz, decreasing down to about 34 Ohms at 10kHz. If the batteries start under 10 Ohms and drop as frequencies increase, it won't be able to catch up to the battery.

The situation is more extreme in the filaments, as a typical alkaline D cell will have 0.1 Ohms of resistance at DC, dropping with frequency. A 0.47uF cap will become helpful at 10mHz, but since the Quickie isn't a ham radio, we don't have to be too concerned way up there.