"But the striking thing to me is just how much smaller the Intel NUC is."
Add the volume of the NUC's hideous and ungainly power supply brick. Then make your judgement on size.
If you've taken a mini apart, you'd see that the easiest way volume could be significantly reduced is by replacing the space for two 3.5" drives with space for one modern NVME gum stick - a trick which the already NUC does thanks to its being developed more recently.
The mini has Thunderbolt and and digital audio in/out. The NUC has neither of these.
If you compare the interior of the mini and the NUC, the NUC does not stand out as an marvel of engineering or miniaturization.
Anyway, whose desk cannot fit a mac mini?