You can certainly run several transformers from one main input connector. In fact the C20 plug at 16 Amps is serious overkill and you could use a C14 which is rated at 10 Amps and is far more common.
A lot of audio amplifiers have two transformers run from one power inlet.
If you want 24 V DC out from a 24 V AC input, that should work because the peak voltage is 1.41 x 24 V = 34 V. From that you will lose the regulator drop out voltage, diode loss, reduction due to the smoothing capacitors discharging and reduction in the transformer voltage due to the current being very peaky. However, you have 10 V so you should be OK.
However, if you put the two 24 V windings in series, there will be over 60 V available, so the voltage drop will be around 36 V. If you want 5 A, you need to dissipate over 150 W of power in the heat sink. I think that would be a poor way of doing it.