On my own finances, plugging my "preferred" numbers into the NYTimes calculator along with a plausible house price and financing that I would buy, changes the rent/buy difference by more than $1.5M over 25 years (!!!!).

You're not doing anything wrong, you're just plugging in some more accurate knowledge about the local housing market that you have. The calculator had to use some sort of assumptions, so they seemed to have gone with medians or averages that made sense at the time.

I tried playing this game too when I bought my house. I ran Monte Carlo simulations which concluded that buying a house was a bad idea, based on historical data. Plus, this whole new "Covid" thing was surely to crash the housing market, right? I ended up buying a house anyway, and found out a little later that my projections were completely wrong. You can't predict the future, after all...