I found evidence of an earlier shotgun start while doing research for my book "From Field to Fairways."
North Oaks Golf Club, a Stanley Thompson design built on James J. Hill's 5,000-acre estate in the northern suburbs of St. Paul, opened for play in 1951. I wrote this, based on information from the club's records: "By July 1953, the club was up to 331 members. Guest day on September 2 was so popular that the pro shop instituted what is thought to be golf’s first-ever shotgun start, in which players took assigned places on each hole around the course, began with the sound of a shotgun blast, and played the holes in order until they returned to where they began."
I can't absolutely vouch for it; I wasn't there. But North Oaks says their first shotgun start was Sept. 2, 1953.