Oh yes, the muscle skinsuit
First, let me address your reductio ad absurdum (which I missed in my previous reply). Unfortunately, that does not work. One of the assumptions in my previous post was that the frictional force was the same for light and heavy wheels. I tried to explain in the "footnote" that that seemed not completely unreasonable to me, as naively the frictional force is proportional to the total weight. A change in wheel weight will be a small relative change in overall weight of rider+bike+wheels, but will have a relatively larger effect on the moment of inertia. Once you send your wheel weight to infinity, the frictional force will blow up as well. So the reasoning about torque etc does not work any more. Well, that and (as you noted) the fact that you would need an infinite amount of energy to get things moving in the first place.
I want to agree that rim weight only matters when there is an acceleration, but I think what I am taking away from this discussion is that (a) my understanding of basic physics is not that great and/or (warning, captain obvious) (b) the "real life" situation of a wheel is very complicated.
So yes, heavy rims require more total energy to get up to a given speed. In the ideal (intro physics textbook) case, once it is going at a constant speed, there is no more work required to keep them going. But we do have all kinds of retarding forces, and as I was trying to think through there might be an advantage to a heavy wheel in how the system of rider+bike+wheels respond to the retarding forces. (And just to repeat, I agree that in almost all cases the disadvantages outweigh the potential advantages.)
And while one guy doing something does not mean that it is a good idea, I would say that more people have similar experiences. After all, Fibre-Lyte brought up the idea in his post, and I have seen all kinds of discussions about how heavy wheels "hold their speed" better. Not that a discussion among cyclists has anything to do with reality, I give you that