The extra resistance (2 to 10 milliohms probably) of the wire between the two boards, together with the smoothing capacitors on the downstream board, acts as a low pass RC filter.
Also the capacitors on the PS board may be low-inductance types, and filter HF noise more effectively than the caps on the amplifier module.
Having the diode bridge further away from the amp module's inputs means the antenna action of the rectifier's leads is less effective: the RF power of over-the-air rectification transients at the inputs is reduced by the square of the distance.
The layout of the amp as shown is good, except that I'd shorten and cable-tie wires to help prevent hum loops.
Get a nice AC-DC converter brick and call it a day. If you want to be fancy, stick a linear regulator on the output.
The ones that I've seen don't put out their claimed current, lose regulation badly at high load, and put a lot of hash on the DC rails. Seriously ugly. Opening them up, the circuit design often looks like it comes from the 1980s. Physical design usually doesn't meet standards for HV separation and creepage.
Rather than a linear regulator, lots of filtering (a few 100nF ceramics and a few millifarads of smoothing) would help. And a high-powered TVS or two for when the power brick dies, as it will.