His error is assuming that all the energy imparted to the boat by wind and sea is ultimately converted to heat by the anchor rode. Although he touches on the correct description when he says "On considering the physics of the matter, the most obvious function of the rode is to store and release kinetic energy." The next line is where he misses the point, "It's least obvious but, I suggest, its most important function is to dissipate kinetic energy as heat."
There will be a very small amount of heat produced by friction in the links or rope fibers that is dissipated to the water. But the primary method of releasing the stored energy is to move the boat back. The catenary in the chain acts like a spring which very efficiently stores and releases energy with little heat created.
Saying the anchor doesn't work is a little misleading. An anchor doesn't "do" work just like a piling or bollard doesn't "do" work. The anchor when functioning properly doesn't move - it works but doesn't "do" work.