That being said, the biggest reasoning mistakes I run across are:
1. "You pay the landlords expenses plus some profit." Not true. The rental market is just that - a market that fluctuates with supply and demand. There are plenty of landlords who are losing money on their rental property.
2. "Once I have paid off my house I'm done paying for housing." Not true - you still have taxes, insurance, and maintenance whose costs will most likely increase over time.
3. Forgetting about the opportunity costs. Great - you paid off your mortgage. Now you have $500k in equity. Guess what - if you took that $500k and put it in a 5 year treasury you can earn a risk free 3% or 15k/year on that money. Better yet stick it in a broad based index fund and you will grow 6% albeit with more risk. That's your opportunity cost of your equity.
Overall this is such an emotional subject for most. I'm personally glad to see some push-back on the "buying is always better" argument because its been dogma for some time.
Edit: fixed typo