What about Parental Controls?
Or do you want to spy on your children's www activity?
There might be other software that logs www activity despite Private Browsing deleting the history.
Another possibility would be to tell us what exactly you want to achieve with deleting the Private Browsing feature, so maybe some users might be able to help you.
Proper help can only be given with proper communication, and you have children, so you should know, that communication is key.