Since you asked
I know Aeliop loves doing this and I hope he chimes in on this thread...
Does anyone else have any information on this or any experiences with it?
Your wish is my command
Before we go any further these are just my thoughts and opinions. I am not saying that I am right and everyone else is wrong nor do I have any scientific case studies to back my theories.
Balance
If someone held a gone to your head and threatened to shoot you in the head if you didn't walk a tightrope or a narrow balance beam with the ground 100 feet under you. With your life depending on it what would come the most natural to you walking with your bare feet over the rope or wearing Nike's? There is no shoe on the planet that replicates the bottom of your feet.
Feel
Shoes cover toes as well. These toes are essential to balance, gripping, and manipulating muscles in the foot. I always grip the floor with my toes and feet, flexing my quads, calves, gastrocnemius. Don't underestimate the importance of not only developing these muscles but using them to give you greater balance and control. Later on when I compete I use these same gripping, balancing, flexing, and controling techniques as stand barefooted on a stage.
Safety
Your foot is a custom made by you for you to balance yourself on. If your body is in balance and then you put it under the stess and durress of heavy weight, well that uneven surface that is a running show might put your body out of balance or alignment
Practical
If you have a nice pair of sneakers or running shoes that you wear to the gym
they are not made for 500LBS+ people. What I mean by that is that if you are 250LBS and lets say you are doing reps with 500LBS. I am just guessing it wouldn't take someone long to ruin all the gell, shock, and springs of a good running shoe.