Very good point because in my country elder care is free. Daily visits by an assistant is on the municipality. We're not a leader in GDP, but we take care of our citizens, whether they be sick or old.
What doesn't show up in GDP numbers are cultures where people take their elderly relatives into their own homes and care for them themselves. Same as hiring someone to look after your kids, or taking them to day care, shows up in GDP; while having their grandparents or other family care for the kids while you're off to work does not.
One person's elderly parents never have to worry about being assisted, while the other can only do so if they can afford it. And you're arguing GDP semantics? There is something seriously wrong and sick with this.
There's a reason why healthcare expenditure per capita in the US is multiple times higher than any other OECD country, and why health insurance companies are some of the biggest public ones.