Dedicated single-purpose hardware with models would be even less energy-intensive. It's theoretically possible to design chips which run neural networks and alike using just resistors (rather than transistors).
Such hardware is not general-purpose, and upgrading the model would not be possible, but there's plenty of use-cases where this is reasonable.
It's theoretically possible but physical "neurons" is a terrible idea. The number of connections between two layers of an FF net is the product of the number of weights in each, so routing makes every other problem a rounding error.
But resistors are, even in theory, heat dissipating devices. Unlike transistors, which can in theory be perfectly on or off (in both cases not dissipating heat).
The thing is that the new models keep coming every day. So it’s economically not feasible to make chips for a single model