Is freeness enough for counting rational points?

Manin's conjecture predicts the number of rational points of bounded height on a Fano variety, but for the predictions to hold, we must first remove a "thin set" consisting of rational points lying on certain special subvarieties and lifting to certain special covering spaces. It might be better if we could instead identify the bad rational points to remove by their intrinsic geometry. It may seem that rational points do not have any intrinsic geometry, but recently Peyre has given two proposals to do this, one measuring the freeness of a point and the other using all the heights. I will explain why the freeness proposal is not, alone, sufficient.