Schottky diodes would make no difference to speak of.
And they have no place in power supply rectifying anyways, that’s not what they are meant for – or good for.
Well, Schottky diodes would have 200mV less voltage drop, resulting in much less energy burned off in the diodes. However, this then obviously shifts to the power regulators, but these are on a heatsink and are currently not as hot as the diodes.
I’m still a puzzled though, because my estimate for the forward current is only about 0.3A. An explanation could be that the reverse current is very high, maybe because of aging. However, I’d not expect this to happen to all of them, or at least not in the same rate, but my finger says they are all about the same temperature.