In general, aggressive screening has been extremely useful: pretty much everyone gets screened for PKU, sickle cell anemia, cystic fibrosis, MCAD, hypothyroidism, and another ~20 conditions at birth.
Funny you should choose neonatal screening as an example in support of your argument. Firstly, the things that *are* screened are a carefully curated list of unambiguous genetic defects that 1) have immediate deleterious effect on the development of the child and 2) have an intervention that will alter the trajectory of the disease. This is why, for example, no one screens the population for Tay Sachs. In fact, the ~30 tests that are part of the standard panel now are all that's left after *hundreds* of candidate disorders were considered for screening and rejected by the American College of Medical Genetics.
Secondly, the positive predictive value of these neonatal screenings is incredibly low. That means that for every true positive result (i.e. a disorder detected correctly) there are 50 to 100 false positive results, which are subsequently ruled out by a different, more specific confirmatory test. However, this same problem (low PPV) is endemic in the mammogram/pap smear/PSA tests, and these result in deleterious effects on many people who have false positive results compared to the number of lives saved from true positive results.
Also, it's worth noting that they still perform the neonatal screening because the human cost of missing a diagnosis in an infant for these *treatable* diseases is so profound, but many parents are put under serious stress due to the false positive results.
Overall medicine is increasing the use of screening, not decreasing it.
Going back to the BRCA example: ask any medical geneticist what is the single most important screening tool. Every one of them will tell you it is "family history". If family history indicates a specific risk, they may counsel further testing such as BRCA. This is different than en masse population genetic screening for "cancer risks".
Recall the original poster was citing cancer risk. Have any examples of a trend toward expanded *cancer* screening? I gave two widely known examples of the trend moving in the opposite direction for cancer screening (in fact, these are probably the two most publicly visible/prevalent types of cancer screening), and correlated this to the issue of laypeople misinterpreting GWAS "cancer risk" data such as was provided by 23andme.
Tangentially, I hope that Theranos actually manages to pull off their plans for microfluidics testing. Look at their projected pricing, and the fact that they want this to be available at every corner Walgreens.