I can understand wanting to practice sports photography, as there is a trick or two to doing it well - and to play devils advocate, kids play more amateur sport than adults, and you need a press pass to take photos at a lot of pro adult venues. That said, I wouldn't randomly rock up at any playing field and start snapping photos of other peoples kids - not in a million years. Its one of those things these days where people are understandably wary and sensitive. I think she's overreacted here, but I'm also not surprised.. and fwiw, I do agree the question of whether she would have accosted a different photographer is valid. She doesn't seem like a very nice person.
In my last job I was asked to take some photos at a kids community triathlon event that our company (a charity) was running and sponsoring, I had all the official event gear on, a pass identifying me as one of the photographers - and I still got queries from a couple of people - who in fairness were happy enough when they realised I was with the company. But I remember being a bit angry at my boss for putting me in that position and felt like it should have been communicated better to participants and parents where we would be stationed etc. I got some great photos though.. little determined faces on the bikes, their elation and pride running across the finish line, stuff like that. An adult event of the same ilk would still have yielded good shots I'm sure, but for kids everything is new, every achievement is that much bigger to them.