One of the big appeals to our sport in getting to interact with the drivers. If we lose that we lose the sport. Big time drivers can make more money off merch than they can racing. Plus, you can blow a motor and still sell merch. That may be the best way to keep things rolling during slow times. Interacting with fans definitely builds a driver's brand and could add longevity to a career. I agree that this should be done before and after the races, not during competition.
One thing that hurts visiting after races are too many support classes. It is tough to ask teams to sit around for an hour and a half after they are done, waiting for the other classes to finish. I know a lot of drivers have some very time-consuming schedules, that is why making sure to schedule time each night is very important to the fans, the track and even the drivers. Lose it, and the sport is doomed.
For the record, I in all my years have going, have never seen a driver who was approached at a decent time, ever ignore a fan. Even with the hard work and travel, most should be glad they are not stuck doing 40 hours a week. Only made possible, by fans buying tickets.