Assuming you are in income/savings bracket where private school tuition is a material expense, how do you think about the two options? To be clear, if you go to private TT, you will not have the funds to exercise option #2 below, it is one or the other.
Send kid to private TT. Odds are not HYPS as those seats are reserved for mega donors, URMs, and prodigies, but odds are good for a Top 20 school. Fantastic education/resources. Potentially building a network for life. Paying $56K (starting, it then goes up each year by 5%) and you donate $5K after-tax each year (you still need your school's support for a Top 20 after all!). Total cost over 13 years is $1.25MM
Send kid to a good public. Kid interacts with the 'real world', still has a shot for a Top 20 school, but may be looking at a good state school or Top 30. Two years after the kid graduates from undergrad, you say *SURPRISE* and write the kid a check for $1.5MM (assumes you invest each K-12 tuition check for a 4% return instead of giving it to the school). Your kid will now have the capital to invest: in real estate, their education, the market, a start-up etc. They could start their own business. Or maybe just take more career risk they would otherwise not be able to. They would probably spend $100K of it on clothes, travel, restaurants and premium booze - but would still have $1MM+ in most reasonable cases.
What is best for the kid? What is best for the parent? What is the better value? What are YOU doing?
p.s. "it depends on the kid" is not an acceptable answer! You can assume the kid is a healthy and normal 5 year old and you don't know what the kid will be like at age 10 let alone 20. But you do wait 2 years post undergrad to give you the option to renege if you think it will all go into drugs etc., as a 'fail safe' and to build some more maturity.