A mortgage is just renting the money instead of the house. In both cases you're paying for the time value of money. It's just that in many (but not all!) cases you can get a much better deal renting the money than renting the house.

Technically correct, but my point is that these costs are frequently ignored and in order to make home ownership more profitable than renting these costs must be amortized over a very long period of time.

No, but it highlights the fact that even when you own a home you're still "throwing money away".

You are not throwing it away. You are exchanging money for a place to live.

As opposed to exchanging it for a place to live, and getting some of that money back when you sell.