Why would paparazzi be able to stalk her at public school but not private? There are lots of celebrities, do they have paparazzi outside their kids' schools
Maybe he prefers that school for his kid, since he's a fighter and she might not have much in common with kids in a private school. Or like said it's the best local school. Maybe there are better ones, like Blue Hal, but they're a long journey away (Blue Hal isn't a long journey for Trunks or Goten).
That Blue Hal one isn't a private school necessarily btw. It just says an elite school. Besides distance or culture fit, it could be Videl didn't get in.