Etymology of phrase "Let's Go <favorite sports team>!"

Usually, "Let's go [team name]!" is used for teams with two syllables in their names and "Go [team name]!" for teams with just the one. They are used to encourage the team to move and go for it, I suppose.

The source of the cheer + clap is usually attributed to a song by The Routers named "Let's go (Pony)" (YouTube) from 1962:

The Routers first release in September 1962 was the guitar-driven instrumental "Let's Go (Pony)",[6] which reached #19 on the Billboard chart. Its infectious “clap clap clap-clap-clap clap-clap-clap-clap Let's Go!" chant became a favorite of cheerleaders and crowds worldwide.

That said, the Chicago White Sox used a song by Captain Stubby and the Buccaneers named "Let's Go, Go-Go White Sox" (YouTube) as far back as 1959:

The song first appeared in 1959 during the White Sox' run for the AL pennant--the team's first league championship since the infamous Black Sox Scandal of 1919. It was written by former White Sox minor leaguer Al Trace and his friend Walter "Li'l Wally" Jagiello.

However, the song did not become famous until 2005.

I expect that the "Let's go [team name]" cheer and other variants came about after the cheer-and-clap from the song by The Routers became popular.