Speaking as a teacher in a private school, you sound a bit daft and deluded.
Most private schools do not have a normal teaching day until 5 or 6pm. They might have a slightly longer school day, but most day schools still finish lessons before 4pm. And yes, the children do get tired, even the secondary aged ones.
Many private schools will offer paid-for wraparound care until as late as 6pm, but so will many state schools. If you've managed to find a private school that offers this for free then good for you, but most charge an additional fee, which is likely to be more than the cost of the equivalent wraparound care in a state school. Like in state school, this will not be additional lessons and will not be run by teachers but by after school care staff rather than qualified teachers. The children will sit around, have a snack and play games and chat or whatever.
Often, if "wraparound care" in private schools is free, then it is in the form of clubs or activities. These are intended for children who have a genuine interest in doing the activity, but sadly there are always a number of unfortunate children who are forced to sign up every day of the week because parents don't want to use paid wraparound care. These poor children are then made to run around a hockey pitch in the rain when they don't even like sport, or doing drama club when they're really shy, or whatever else. I feel really sorry for these children, and wish that their parents would just pick them up from school, or sign them up to the paid club where they can just sit around and chill.
If you're able to afford private school for your child then that's fab, they will get access to a range of different activities which probably wouldn't be available in a state school, and they might do a bit better in their SATS or whatever. But if you're making decisions about where to send your child based on the finish time of the school day then that's really daft. Many working parents send their child to state school and pay for wraparound care, which is obviously much cheaper than paying private school fees.