Plus the smaller chips (like the CPU) are designed for lower voltage and current. They can't handle dialing up the power, they'll melt.

I recently got a have Mini-PC which a processor with a TDP of 6W and it uses run of the mill SODIMMS and the power supply for that stuff is a pretty regular wall socket power adapter, the same kind you would see for, say, a media box.

I suspect it's not even a few watts (at 3.3V 1W is around 300mA is quite an insane amount of current for a signal line), more like tens or even hundreths of a watt.

Mind you, what really changes here is voltage rather than current: these things run at a lower voltage, which helps with speed and in reducing the power dissipating as heat (so they waste less power and heat up less) and that's were signal integrity on longer signal traces becomes more of a problem because lower voltage signals are closer to the noise level the drop in voltage from the resistance of the circuit board lines because a higher proportion of the original voltage so the longer the trace the more likely it is that whatever reaches the other side is pretty much at the same level as noise.

Still matches what you wrote, by the way, as power = voltage * current, so all else being the same lower voltage does mean less power consumed. It's just that you were a bit off on the scale of the power consumption involved plus there's some more stuff related to using a lower voltage not just for lower power dissipation but also lower heat generation (which is directly derived from lower power dissipation) and higher speeds (which is for different reasons).