The SSN was never intended to be a secret. Nobody in 1938 thought that one day banks and other businesses would use it as a credential to establish identity.
For GP, actually yes, people did think it would be used as a credential to establish identity, and many feared (correctly, as it turns out) that would happen. The phrase used by those people was "Papers, please!" with a German accent, since it was well known by the mid-40s that identity papers were used by the Nazis to control the population. Later variations by people my age was to use a Russian accent since the Bolsheviks also used papers to control the population.
My uncle and aunt never had SSNs for that reason (and no, never paid into or collected social security either.)
I have a friend who is now in his late 50s who has never given his SSN to anyone except the IRS and his employer -- which is actually the law, or at least used to be, perhaps it has changed.
Rather than assume older people are less privacy conscious, the correct assumption is that younger people are less inclined to protect their privacy. Many millennials don't even believe in privacy. I blame it on the generally poor education in history that most younger people have received.
If it was a law, it has since changed. The IRS has the W-9 form specifically for the requesting of an SSN (or EIN if a company). At my job I require it before issuing a non-employment related payment that will get reported to the IRS on a 1099-MISC.