@Joe Splitrock I would tend to agree that in most circumstances that buying is better than renting, however @Rich Chen's situation does not prove to be so.
1) He would have to sell or cash out refi his rental, essentially eliminating his cashflow. If he sells, he likely will pay significant closing costs (commission, title etc) and taxes (capital gains).
2)Even after he gets the cash, it may not be enough for the down payment so he would have to come out of pocket some cash.
3)Even after losing his cashflow and coming out of pocket, his PITI is almost the same as what the rent would be.
4)His equity would be trapped again in his house vs a rental, net the same except he will have less equity (see point 1)
Given this, how is it better to buy? The 3K he is saving by renting Paying 2K in rent vs 5K in PITI, he can put towards investments and possibly a better quality of life. If they did lose their job, could they survive longer paying 5K a month or 2K? Besides, there are costs to home ownership outside of just PITI.
It totally makes sense where I invest in Ohio to buy vs rent from me. The PITI would be almost 1/2 of what rent is, so it makes a lot of sense. In these high priced markets, I would wager that the benefits are not the same.
You suggest essentially a house hack or to move into a part of the city where it's more affordable, however the OP stated he wanted to be in the better school district and have more space for his family. I'm simply offering a suggestion that allows him to achieve both with less risk, IMO.