So in context of mortgages being bad, you're saying that house prices would be lower if only people who have the money saved up or inherited to buy them are bidding on houses? Wouldn't that make supply much lower?

If the mortgage did not exist, and a significant portion of buyers (or the sellers) did not have access to same/similar amounts of capital, many people would probably restructure their financial lives in order to facilitate house buying without a mortgage. The same/similar number of people need housing and move, upsize or downsize, so the nature of the supply might change but probably not supply in general.

If the mortgage were to suddenly disappear I would expect the housing market to have a similar contraction as in the recent recession when poorly understood risks from mortgage backed securities caused a decrease in mortgage availability.

But how much would it contract when you can rent the properties out? Rental and purchase prices are linked - the price floor of a house is related to how much you can rent it out for if you sold it to a landlord (or kept it yourself to rent to people). People would stop buying houses and start renting. The demand for renting would go up, and for buying would go down.