Yes, this is broadly correct. The free market will (roughly) arbitrage out any differences between owning and renting. The hidden factor is that whatever money you have in house equity represents opportunity cost that it isn't in investments. If you have 400k in a house and the stock market returns 6% over inflation, then the opportunity cost is 2k per month in interest, which is comparable to what you'd pay in rent.

There are tax advantages that favor owning (in the US), for a primary resident and not an arbitrageur - mortgage interest and capital gains when you sell are not taxed, while capital gains in a non-retirement account are.

You can gain by appreciation and leverage, of course - but you can just as easily not, you don't know if your city is going to be the next high-flying Austin or Boulder, or run-down Detroit. My own house has been flat in estimated value for four years in an area that I thought would continue to rise.