No you're not. You should be able to afford many years of rent with the cash from selling a house.
A paid off house gives you a type of stability, belonging and security that rentals don't generally offer (at the expense of flexibility and mobility). I'm not trying to say that owning is better than renting, just that they are different enough for the "buying your house isn't an investment" statement to be mostly true.
True, but calling renters "homeless" would be ridiculous.
Unfortunately I think people have forgotten what an investment really is. An investment is just a capital allocation. It's not about buying now and then selling later for profit. That's called speculation. A good investment will continue to deliver value over time. A house literally keeps you alive. Of course it's an investment.