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Now we're talking, except you'll just encourage companies to form to pool cash to buy houses then redistribute rental income. The only real solution is to ban sales of non primary residences.
The countries who forbid mortgages on 2nd homes tend to back it up with laws that give tenants more rights and rent control. As you suggest, the downside is that a small number of people hoard most of the property but the upside is that its not treated as a get rich quick scheme by landlords and tenants receive stability, security and lower rents in return.
I don't think it would be a bad thing if the UK started to chip away at the obsession with property ownership by making renting more attractive.
The one change I'd love to see is banning mortages for any residential property that isn't your primary residence.