Thanks for the note. I am in fact quite careful with the tubes and stems, because they're so delicate. And I just went through two new tires that both came with removable but defective Presta valve cores. So I'll put those in the bin and hope that I come across some cores, which is not unthinkable. I'd order something, but have to wait until I have more stuff to order.
Perhaps part of the problem is a decline in tube quality, as the good suppliers are battered by the cheap ones. Like the old saying in economics: "Bad money drives out good money."
I think it's an actual design issue. The Schrader valve simply has a larger base, thus there's more surface for the rubber and metal to stick together, and less stress on the joint. This may be one of those cases where a technology that's good for technical and sportive cycling is not the best for utilitarian cycling.