I'm pretty sure its just human nature. Families have been important through out history. Spending your last days surrounded by family was often something people sought after, which is why when groups of people (based on religion or government or tribal rules or whatever) prohibit families from burying their dead, its an effective way to punish those families.
Spending time with the family unit, how you interact with that unit, choose to honor it or follow the unit's codes of conduct are all topics that have been around as long as humankind has. The United Nations states, "the family is the basic unit of society and should be protected by appropriate legislation and policy" and go on to state, "In spite of the almost universally recognized roles attributed to the family, it is important to acknowledge that there are numerous forms in which families are organized. Such variety is concomitant with the multiplicity of forms of social organization, and cultural and religious values."(http://www.un.org/popin/icpd/prepcomm/official/rap/RAP4.html)
There was a really interesting academic paper put out in 2016:
https://www.mdpi.com/2077-1444/7/5/53/pdf
It talks about religious traditions concerning end of life (when someone is terminally ill). Most of these religions/cultures going back to the beginning of time, often have the one who is going to pass away with their family in some way. In western secular cultures we still have many of these traditions. Usually patients in hospice want their family to come and say goodbye (or the family wants to say goodbye). In many ways, being able to get closure is really important to most families. Universally speaking, dying alone and forgotten is considered sad, although maybe not morally wrong.
I know that in the US and some places abroad, when you leave your family unit (due to abuse, political disagreements, religious disagreements), you are likely not going to follow the traditions of your culture/family. But that doesn't mean the cultural pull to make peace with your family isn't there. I think this sort of family and inner conflict is really interesting and makes for good story telling. If you characters don't have a nuclear family, how do they come to terms with that at the end of their life?
Back to the OP- you asked if we would spend our last days with our families- My answer is absolutely. I am married and have little kids who I love more than life itself. So if I was terminally ill, I'd want to spend as much time as I possibly could with my children and my husband before the end.