If you cut rent, then how will people who do not own their homes have a roof over their head?
If you pay for people's mortgages, then why would anyone work when the government can just literally buy them a house? That they can sell, or pass to their children.
Home owners are not being penalised. They can get help with the interest, which is that will keep the bank off their back. Long term claimants such as those unable to work can and do lose their houses. But most people with a mortgage claim UC when in-between jobs and end up back on their feet again. Benefits are not a reward for doing "the right thing" such as buying a home, they are a safety net.
People who rent are not doing it on purpose to claim benefits. They are people who can not afford to buy. Even if you have your rent paid, if you are renting in the private sector, you are not in secure housing.