Indeed, obituaries are public things (in the US, they are mostly paid additions to newspapers or websites, If you see death notices, which often contain only facts like the death date or the funeral details, they are often printed as a service to the community). Usually the deceased, the family or someone representing the family (like a funeral director) will write the obituary and have it published to honor the person's life.
I almost always remove the names of living when I use an obituary. Sometimes when I know it won't be visible due to a privacy level, I leave the names in. This is to respect the privacy of the living persons. (I see it becoming more common for obituaries to leave out names of living family members and say something like "survived by a son and two daughters.") Anyway, sure the names are "out there," but it is a WikiTree policy not to add profiles of living persons without their permission. (Please see:https://www.wikitree.com/wiki/Help:Living_People).
Also, using Find a Grave and only Find a Grave as a source is not a great idea. There are many errors in that site, as well as "burial unknowns" etc. (Which, to me, defeats the purpose of finding a grave! If there's no known grave, why add it..but I digress...) Birth records..VOILA. Yes, use those by all means. And use Find a Grave, but not only Find a Grave, if that's at all possible.