the PSU doesn't sit there and think 'I'll sip the power supply to recharge and hope there isn't a drum beat anytime soon', it tries to recharge its capacitors as quickly as it can (made worse by the rectifier bridge introducing a voltage drop, which increases the spikes of current drawn).
The current that an amplifier uses to drive loudspeakers comes from the charge held in the reservoir capacitors; the capacitors are (and can only be) charged at 2 x mains frequency (full rectification).
The DC side of a power supply is isolated from the AC by the reservoir of charge in the capacitors; when you design a power supply, you calculate the required capacitance for the voltage and current requirements, if they are inadequate, you will get intermodulation distortion.