Beyond refusing the inheritance, could this be avoided by leaving the inheritance to the state in a will? In that case, would the state inherit the debt?
I have experience with government run elder care programs and the (privately run) estate planning that goes with it. Shoehorning all your worth into a trust (there's more to it than that) is something that not so affluent people commonly have to do in order to qualify for state run elder assistance programs/subsidies that they don't qualify for and couldn't afford themselves (fixed income) because their money is tied up in a non-liquid asset (their house). This isn't something just for the affluent. Basically if you own your own house or have a lot of equity in it it's what you have to do unless you want to give the state your house in exchange for a free van ride to the senior center or 7hr/wk of in-home care.
And before you assume everywhere is like SV and that owning your own house makes you affluent by default I would like to state that my experience is on the other side of the country.