Very old machine did some analog processing. Then it switched to digital on dedicated hardware. New machines can do all the processing in software, running on conventional hardware.
There is all kind of fun stuff that you can do with ultrasound, and some are similar to 'synthetic aperture radar'. However, 32 channel is still quite low. Lots of probes for humans are commonly using ~100 channels.
I would expect that they are looking at the return as an amplitude modulated and phase-shifted signal at the carrier frequency. They remove the carrier frequency and then look at the (much slower) modulation frequency.
I'm not sure what you gain from looking directly at the carrier frequency...
The relative bandwidth in ultasound is immense conpared to radar etc where a relatively narrow bandwidth is used. As an example one of our probes in university days was something like from 5 to 30MHz. Doesn't save much if we would mix it down that 5. Instead of 60MHz ADC we would need 50MHz. Not worth it.
I wonder if there's an equivalence to 'synthetic aperture radar' on Ultrasound. then maybe you don't need 32 channels