PET isn’t really suited for a screening population due to relatively low availability and high cost (esp. compared to a mammography). Resolution is also bad and it tends to also light up inflammations. So it probably wouldn’t help too much for reducing false positives.
In the case of patients with dense breasts, which in the article undergo MRI, an argument can be made for PET in both availability and cost now that these patients are in MRI territory. Also, organ-specific PET is working on the resolution problem but I'd agree that if you were to put a patient into a run of the mill PET scanner there wouldn't be much clinical usefulness.