Is the family considered responsible for the mother's death if they refused a tracheostomy for her, preferring another attempt to wean her off the ventilator, knowing that the resources for her care after the operation were unavailable?

You should pray frequently for your sick mother, for Allah answers the distressed and removes evil. Her care is obligatory upon all her children, male and female. The married woman must contribute to her mother's care without neglecting her husband's rights. There is no objection to requesting another treatment method based on the advice of doctors, and there is no sin upon you if death occurs after making this medical decision.