Which is exactly why they might be OK with any product that told people "you might have a problem, go to a doctor" even if the true thing detected by the product was "you have a problem, period". If a product turns out to cause many people to believe they have a problem without going to a doctor, the FDA would attempt to ban it.
Yes, I think there a way to manage the situation that the FDA approves the product.
In the long run it's possible to actually run the studies that are required to diagnose certain illnesses.
These studies cost tens or hundreds of millions of dollars, take months or years to run, and then the process of getting the FDA to approve your product takes more millions of dollars and more years. At the end you have approval for a particular product for a particular purpose, and if you want to launch v2.0 or to use the product's measurements in a new kind of analysis for a new disease, you need to go through the approval process again.
This is entirely incompatible with quickly evolving technology in a market with many small / new players trying out var...
Apple's iPhone 7 Plus decided to add another lense to be able to make better pictures. Meanwhile Walabot who started with wanting to build a breast cancer detection technology released a 600$ device that can look 10cm into walls. Thermal imaging also got cheaper.
I think it would be possible to build a 1500$ device that could combine those technologies and also add a laser that can shift color. A device like this could bring medicine forward a lot.
A lot of area's besides medicine could likely also profit from a relatively cheap 3D scanner that can look inside objects.
Developing it would require Musk-level capital investments but I think it would advance medicine a lot if a company would both provide the hardware and develop software to make the best job possible at body scanning.