But, since people don't mine uranium randomly, one may assume that your back yard doesn't have enough to be price-competitive with modern mines.

Of course. The point is that if a nation wants to make nuclear weapons, keeping them away from uranium is a nonstarter. Uranium is everywhere, if you're willing to pay to extract it (in this case, by piling up the dirt and leaching the uranium using a bicarbonate solution).

Fun fact: if the uranium and thorium in an average crustal rock were totally fissioned (using breeder reactors), the rock would produce about 20x the heat energy of burning the same mass of coal.