I'm not sure that anyone cares enough to actually have studied this with any sort of scientific accuracy. So basically it's heresay. "People say..."

But I've been riding fixed for 20 years or so and have done a couple of fixed centuries. The first 10 were mostly brakeless. As I got older I found the stress from climbing and resisting the pedals for slowing and stopping were causing other parts of my body to tense up and ache, not my knees so much. So I put a brake on and I do most of my longer training rides on a geared bike and use my fixed gear for commuting and fun shorter rides. There's some truth to the saying "geared bikes are for people over 40"