In the US, debts of the deceased are paid out of the estate. If there isn’t enough money in the estate, the debts just go unpaid—descendants or other survivors are not responsible.
In your scenario, assuming he owned the residence, it would be sold and the debts repaid from the proceeds; likewise with any other assets he left behind.
Sekiro is far and away my favorite FromSoft game. I think people bounce off because they expect it to be more like the classic souls experience than it is; there are three main differences IMO that you need to know to have a good time: