"Clinicians should be aware that meditation programs can result in small to moderate reductions of multiple negative dimensions of psychological stress. Thus, clinicians should be prepared to talk with their patients about the role that a meditation program could have in addressing psychological stress. Stronger study designs are needed to determine the effects of meditation programs in improving the positive dimensions of mental health and stress-related behavior."
This directly contradicts the GP's claim that meditation is ineffective, and supports the OP's claim that meditation is a good way to deal with "psychological worries".
Also, meditation research has come a long way since the studies cited in your source. Something I dug up quickly: http://www.patheos.com/blogs/americanbuddhist/2014/01/the-ef...