My tree was set to private the moment I begun it because I knew that many of my cousins would not like their ancestry in the public domain. Unfortunately, there are several sites on the web about my family which reveal personal details that I find uncomfortable - I hasten to add that there is no one in my ancestry to be ashamed of, and much of the information is readily available in the newspapers or documented on TV but I fail to see the need for others to parade them quite so publicly, especially when said sites reveal the living. Fortunately, my own branch has remained off the radar, and fingers-crossed, they remain so.
The password for a private tree could be included in a will or tree could be downloaded as a gedcom file and transferred to a flash drive and given to a relative.
I do, however, understand those who choose to have public trees.