I don’t run, but I live as much of my life as possible barefoot. It’s made much easier by working from home, so I can spend a big chunk of the day either in the house or out on the deck in the back garden mooching about barefoot. I tend to keep a pair of flip flops in the car, so if I need to head into town with the car, I can drive barefoot and slip them on when I’m heading into the shops and so on.
I much prefer it, and I find I have far fewer problems with my feet than when I worked in an office in trail shoes most days. I used to get really bad sweat rash on my feet, even though I washed them daily, changed my socks and so on.
I have absolutely no empirical evidence other than reflecting on my own experience, but I’ve noticed that when I drive long distances, I stay more alert for longer if I take my socks and shoes off and drive with my bare feet on the pedals. I guess there’s something stimulating about the tactile experience of having the soles of my feet on the textured rubber.