what's your timeline on libraries? i guess you're counting ancient, private collections. public libraries happened well after copyright was established and just had to go to court and make their case. first sale doctrine indeed saved the day.
the more interesting case for me is that xerox was allowed to exist, and libraries fought successfully to allow their patrons to use xerox machines within the library (1973 Williams & Wilkins Co. v United States). this freedom may not have been established had it been any other circumstance than a medical journal suing the medical doctors xerox'ing the papers for their own research. the public attitude was "bro, lives are on the line here, let the doctors make copies" and we got the four factors of fair use outlined in the 1976 Copyright Act
If you do not consider the ancient, famous libraries to be public, Wikipedia puts ~1600 as a first date of modern public libraries. While copyright has a first occurance date in the 1700s, in some locales much later.
Either way it seems public libraries were around at least a century before copyright, but by other measure, in some places, much longer.
the more interesting case for me is that xerox was allowed to exist, and libraries fought successfully to allow their patrons to use xerox machines within the library (1973 Williams & Wilkins Co. v United States). this freedom may not have been established had it been any other circumstance than a medical journal suing the medical doctors xerox'ing the papers for their own research. the public attitude was "bro, lives are on the line here, let the doctors make copies" and we got the four factors of fair use outlined in the 1976 Copyright Act