Seems to me like two different questions:
(1) Should an average person try to study medicine and statistics and find out the answers for themselves, or use an expert opinion?
(2) Are people recognized by current laws / credential systems as "experts" really the best available experts?
I would support people getting information about their health from organizations like 23andMe. But I don't expect 23andMe to do all research on their own -- at some moment they are going to rely on some peer-reviewed study. There should be a system where the studies are not easily gamed by financial interests of pharmaceutical companies, or by pressure on scientists to publish even when there is nothing worth publishing.
Should an average person try to study medicine and statistics
An average person should not do many things. That is not a good reason for people who are not average to abstain from these things.
John Ioannidis has written a very insightful and entertaining article about the current state of the movement which calls itself "Evidence-Based Medicine". The paper is available ahead of print at http://www.jclinepi.com/article/S0895-4356(16)00147-5/pdf.
As far as I can tell there is currently no paywall, that may change later, send me an e-mail if you are unable to access it.
Retractionwatch interviews John about the paper here: http://retractionwatch.com/2016/03/16/evidence-based-medicine-has-been-hijacked-a-confession-from-john-ioannidis/
(Full disclosure: John Ioannidis is a co-director of the Meta-Research Innovation Center at Stanford (METRICS), where I am an employee. I am posting this not in an effort to promote METRICS, but because I believe the links will be of interest to the community)