Taxing property doesn’t really work, because landlords just pass those taxes on to their tenants.
Perhaps. But the renting business would get way more risky, since you need to pay those X% of the value whether you have a vacancy or not. I expect this would disincentivize the rental providers and would a) stop buying and b) sell a lot of their properties, both leading to a significant drop of the prices on the market.
Most people are renting housing not because they like the lifestyle, but because they can’t afford to buy their own. If you make buying housing affordable (mortgage is what they pay for rent), then the actual number of people who really need to rent as opposed to own is pretty small and could be covered by some kind of social/city provided housing.