My wife is about due with our first child and we are finishing renovations to our primary residence in the mean time.

:) That would’ve been perfect if I were single, but we have a newborn so I wouldn’t feel comfortable with having a roommate and sharing common areas with someone else at this time.  

I am 33 years old, my wife is 31, and we will be welcoming our first child into the world in early/mid January.

I had a sick child so my husband closed in my absence, so this is all being relayed second hand, but here goes.  

So even if the child did all the downpayment and the loan was only in the fathers name?

If you have flexibility on a location there is almost always a home for sale for a similar price with a better assigned school (particularly now with the population aging/retiring and both parents of a child more likely to work).

My wife and I are in a transition period -- one in which I retired my decade-long business, became a stay-at-home dad for our 2nd child, and am now doing RE investing fulltime (managing my local properties and making good progress on remote-investing in other parts of the country).  

Counselors believe that it is "victimless crime" , nevermind a child cannot defend themselves.

I was married by 24 with a child and my wife didn't work.

Your analysis is child like.