Housing prices would rise even if money wasn't being printed. Because when I buy a house, the money doesn't disappear, it goes to the seller.

Imagine there was only one bank, we'll call it Banky Mc Investment Inc. or BMI.

Fred wants to buy a house. That house "costs" $1 trillion dollars. Bob owns the house and decides to sell.

Fred goes to BMI and says, "can I have a loan for $1 trillion?" BMI says "sure thing Fred" and BMI transfers "$1 trillion" to Bob's account. BMI then goes to Bob, and says "hey Bob, it looks like you've got a lot of cash in your account. Would you like to invest it in the BMI morgage investment pool, ultra low risk, backed by valuable real-estate? Bob says "sure thing!" and BMI's books balance. Sure, Fred now has a lot of debt, but there is no reason why this couldn't happen, even with a "limited money supply".

You don't need cash to put IOUs on the books.