I feel it's a net win anyway, if the procedure is not invasive and cheap, you could take one every so often (yearly?) and if something shows up you just double check with a better method or w/e. Better than nothing, which is the current alternative.
A couple thoughts -
1) If this is the only/main objection, then the test is good news for those that are analytically capable, even if it's potentially bad for Joe Median.
2) While what you describe is how it probably happens in general in the USA, but I wonder if it plays out differently in, say, the UK / Canada under NHS/Medicare. In those systems you don't get to pick your treatments; your doctor will only prescribe a follow-up if it meets ROI standards. (Then the health system picks up the bill). Of course you can go private but there's a pretty big barrier there, and perhaps having your GP push back more strongly (as they would if you were not actually at high risk) that might prevent many people from inferring too much from a potentially-false positive result.
From https://www.ama-assn.org/sites/ama-assn.org/files/corp/media...
"Screening mammography reduces mortality from breast cancer, including in women younger than age 50 years. However, screening mammography carries harms such as false positive results that can lead to additional imaging and invasive biopsy procedures, and overdiagnosis that could lead to treatment in patients who may not benefit from it. The USPSTF considered the balance of benefits and harms using a commissioned targeted systematic evidence review of randomized clinical trials and a decision analysis that compared the expected health outcomes of starting and ending mammography at different ages and using annual and biennial screening strategies; it concluded (in part) that routine screening should begin at age 50 years and continue biennially until age 74 years."
Nah, there's plenty of other things you'd try before the biopsy.