George Bernard Shaw was a man of many talents, but he did not say “We don’t stop playing because we grow old, we grow old because we stop playing.” Shaw died in 1950, the quotation does not show up in Google Books until decades after that. When it does show up it is credited to Satchel Paige. I cannot verify that these are Piage’s words, but it certainly sounds more like something he would have said.
Is it important to attribute quotations to their correct source? I think so, but more than that I reject this tendency to reduce people to pithy sayings and not wrestle with their actual contributions.