In addition, many diseases like STD/STIs or mental illness carry a stigma. Information like that, if inadequately protected, can have a social or even financial cost like losing your job.
I don't think changing society to not be embarrassed about things being made public is a good solution to privacy concerns.
Also, we might have a million, non-embarrassment reasons to want to keep things private. I might not want people to know about my cancer because I want to be able to live my life without having to answer questions about it, or have people treat me as delicate. I might not want people to know I am pregnant because I don't want to have to tell them when I have a miscarriage.
Most importantly, I want to be able to have whatever arbitrary reasons I want to keep things private.