Senator
Thinks s/he gets paid by the post
The hospital has no way of knowing whether the estate is $6,000 or $60 or $6 million... all they know is that the estate owes them $6,000 (or so they think but it sounds like they are probably wrong on that). If you send them the death certificate it gives them no more information than they had before.
It gives them knowledge that their debtor is no longer producing income, and to chase the person is likely to result in no money. They will hesitate to throw good money after bad. With person that is still alive, they can get a judgement for little to no cost.
If they want to file a claim against the estate, they only have a limited time to do so, and that is likely to fail. Any account with a beneficiary is not required to go through probate. A probate claim is their only chance, or trying to convince someone to pay that may feel some guilt.
Even if they had a right to collect, they will not chase $6K from someone that was in a nursing home and likely has no assets.
These bills are on auto-pilot from the billing department. They need something to charge the bills off. A death certificate is what they need.