"...but many of the regulations that come down the pipeline are pushed by brick-and-mortar competitors who want to keep competition at a minimum."
Or perhaps the brick-and-mortar competitors simply want for the competition to be on even ground. If the kids were required jump through the same hoops as the legit stands, then the kids likely need to sell lemonade at higher prices than a quarter.
"Police said the girls needed a business license, a peddler’s permit, and a food permit to operate the stand, which cost $50 per day or $180 per year each, sums that would quickly cut into any possible profit-margin."
Correct. These fees cut into profit margins for all stands. Why should kids get special treatment?
"Who stands to lose from a couple of six-year-olds selling lemonade?"
Any purchase I make at one stand is a lost opportunity for another stand. Again, why should kids receive special treatment?
"Kids have been selling lemonade for decades without permits of any sort. They often set the stands up just for fun, but many lemonade stands (or bake sales) are used to raise money for schools, cancer, or sick pets. Lemonade stands represent the most innocent, optimistic side of capitalism out there."
Lemonade stands for kids is a great learning experience. Having to deal with the law is part of that learning experience. A lemonade stand run by kids and shut down by the police is as good a learning experience as one that is able to run without interference from the government. If the startup costs of a lemonade stand are too high, then perhaps the kids should try a different line of business.