Oh yeah! I have run plenty of errands both before and after the barn, and I've gone to dinner with friends post-riding. I'm too old to care what people think. And let's face it: Even on my worst day, everyone on People of Walmart has me beat. However, I will go home first in extreme situations: Got pooped on, fell off into mud, unfortunate sweat stains, bareback riding during shedding season, etc.
For those that find it gross going to eat while that dirty... I will ALWAYS stop off in the bathroom and wash my hands prior to sitting down (and arms if wearing short sleeves). If necessary, I'm not against washing my face either. I figure, I bring water to the barn all the time, and who knows what gets stuck to that straw! So my food at a restaurant is no dirtier than that, and I will never go out if I'm stinky enough to offend nearby diners.