There are forms of autism on the spectrum that would certainly allow a child to reach five, or six, or seven, without a parent or a casual educator realizing it. Many indications are indeed similair to add and adhd. BUT, only a qualified education specialist can make any determination. Unfortunately, private schools can frequently make their own rules, but often (and it depends on the laws in your state), the public school system is required by law to provide an evaluation at your request, even if your child is not enrolled in the system. I would contact your school system's special education dept. or Child Find to find out the exact process.
Good luck! You are your child's best advocate--if something feels funny, push back. Your instincts are probably right.