"Demand implies someone wants somewhere to live, not they they can afford to purchases the property."
No, Q (Demand) requires the buyer be able to purchase the property, meaning he/she can afford it. Otherwise, he/she is by definition NOT in the market to buy at that price and is therefore NOT part of the demand (he/she is off the curve). Can't pay, can't play, that simple. Renting simply means they're playing a different game, taking them out of the buyer's game.
Let me put it another way. Why are prices going up unless people are willing to buy the houses AT those prices? If it's not the hoi polloi, WHO is buying up the houses at the increased prices that the sellers are able to command?