To be even more proper, you'd need to allow the stock market investor to use leverage, just like the guy with the mortgage does.
Or, if you want to avoid the mortgage/leverage complication, we can compare someone buying a house outright with someone investing the same amount in stocks, and uses the returns to pay rent.
> Around here, houses appreciate in value. Quite significantly, actually. A house can gain 100k in value in just a few years.
Total returns on eg the S&P500 over the last few decades have been pretty good, too. And it's a much more diversified and liquid investment than a single house in a single location.
For some people in some places, buying a house might be better than buying stocks, for some others it might be worse. Results also depend on taxes and jurisdiction and personal preferences. But it's not automatic that buying a house is better than renting.