Large data centers are dramatically more efficient at utilizing both the power and the hardware.

At it's core, running a top-tier LLM means linear load of trillion+ weights from RAM for every token. The cost of this operation absolutely dwarfs the cost of doing the actual math. If you are only serving a single user, you can only amortize this over a handful of token streams, times like 2-4 for speculative decoding. If you are running in a datacenter, you can batch thousands of streams, and get correspondingly much more tokens out of the same expenditure of energy and hardware.

Be slightly careful with this. Large datacenters use power to operate the facility (this is measured as PUE > 1). Datacenter-oriented hardware often contains low-diameter, high static pressure fans that consume quite impressive amounts of power that are not counted as overhead in the PUE calculation.

A home- or office-oriented workstation will have much less power-hungry fans (go compare a Noctua datasheet to a server fan datasheet - the Noctua wipes the floor with a server fan if measured by CFM/W, and the internal temperature rise is mostly controlled by airflow and not by static pressure). And in some climates and times of year the marginal facility HVAC cost associated with running a computer may be zero or even negative.

And lower utilization hardware may run in on a more efficient point on the power-vs-performance curve. [0]

That being said, your overall point is correct.

[0] For a dramatic demonstration of how much this curve matters, you can run a non-CPU-saturating, non-latency-sensitive workload on various Intel CPUs with and without turbo on. Seeing a 2x or greater difference in reported power is quite possible. (Intel is terribly at writing platform firmware and the default behavior may be wildly suboptimal.)