Not everyone has lost their jobs. Folks that still have an income should be paying their mortgages/rent on schedule. Those who can’t can defer. This way there’s a reduction in income but not a total collapse.
You'd be a fool not to defer, if the deferral is at almost no cost. We have seen that people will take the financial path of least resistance (as happened with non-recourse loans in the financial crisis), so I wouldn't depend on payment by those who can.
Means testing would cause more problems than it can solve. A somewhat fair compromise would be to increase the amount owed a tiny bit to encourage those who can pay to continue paying.
