If disadvantaged, underserved youth are less likely than advantaged, underserved youth to receive mental health treatment from another provider, then would turning away a child with private health insurance in order to see a child without health insurance be more likely to ultimately lead to a good outcome for both children than turning away a child without private health insurance in order to see a child with private insurance?
Only if the advantaged underserved child eventually gets served. Hence the problem. It's like this (forgive the long analogy):
Imagine you're the only waiter in a busy restaurant. Everyone else is home sick. You've got a restaurant full of tables (45% Rich, 45% Poor, and 10% clearly malnourished and starving), what do you do? Do you try to serve them all, and provide terrible service to everyone? Do you only serve best-dressed, obviously wealthy tables? Do you only serve the sloppily dressed, obviously poorer people? Do you serve the starving people first? What if they can't pay?
Some people try to serve everyone, and either work themselves to death, burn out, or fail miserably. There might be a rare one or two who are especially gifted and talented and succeed. Like the amazing waiter who could handle this restaurant scenario well.
Some people only serve the rich people, because they reason that they can't serve everyone, so they prioritize the tables that will bring home the most tips. Since you're going to be serving, say 50 tables that night, might as well make it the best 50, and ignore everyone else, since they weren't going to tip you well anyways.
Some people will serve the starving people first, because they clearly are in trouble. Others will tell them to go to the soup kitchen down the street, since they can't even pay their bill, let alone tip you.
A few people would serve the poor, but not starving, people first, because they feel wronged by the wealthy and want to give the finger to the man. This group is probably the smallest, because they a) have some clear issues, and b) will soon be poor and starving themselves.
Finally (and this is where I fit in), some people will try and focus on a practical mix of the above, asking the hostess to limit the number of available tables, and put people on a waiting list. The waiting list will be so long that we would advise the starving people to look elsewhere (possibly the soup kitchen) while they're waiting, but that we would buzz them when their table was ready if they didn't find anything else in the mean time. You may even have the chef feed a couple of the most starving people out the back door.
Ultimately, everyone will pick the option that appeals most to them. A lot of people will pick the only wealthy option, but I should also point out that many self-pay customers are not only wealthy. Read up on the Access Healthcare model of Family Practice, where a doc in North Carolina opened a Fee-For-Service office and was surprised to find that 50% of his patients had no insurance or were poor. These people preferred the cash model because the prices were clearly listed and they could budget ahead. They knew exactly what service would cost, and it wasn't exorbitant. Food for thought.