On average a person works 40 hours a week, but "round-the-clock in-home care" means 24x7=168 hours a week of dedicated labor, more than 4x of the typical 40-hour work week. IOW, it takes four people's labor to provide for one person's "round-the-clock in-home care." Of course it's expensive!
At scale, few societies can ever afford this with human labor.
It will be a long while before robots become advanced enough to be competent caregivers. May take longer still to transfer our consciousness into machines so we don't have to age in decrepit biological bodies.
Interim solutions are centralization/statistical multiplexing like nursing homes and having most people dying quickly as to not need round-the-clock care for long once they started needing "round-the-clock care.
Centralization/statistical multiplexing means giving up "in-home" care. The "die quickly" option also has its shortcomings. From a public policy perspective, support legal euthanasia to make the "die quickly" option more palatable, especially if you want such on option to preserve some dignity.