How do you know it goes to a mortgage? The property could be owned by a retired couple and the rent is the income they live on, or a charity and the rent is how they fund their operations.
Or it could be owned by a regular old smelly landlord who just takes the rent money and invests it in the S&P 500 -- which, if you take it away multiplied by millions of landlords, does a Bad Thing to the stock market right when you want the opposite of that.
Sometimes rent includes utilities. What does the landlord pay that with?
I imagine you also expect to waive property tax or the landlords would have nothing to pay that with either, but then what are you using to fund local governments?
You can't just delete an industry and expect it to have no untoward effect.