During a 13-year baseball career, Mincher batted .249, hit 200 home runs, and collected 643 runs batted in. He was elected to the American League All-Star team twice (1967 and 1969). As one of two representatives for the Seattle Pilots in 1969 (their only season in existence before they became the Milwaukee Brewers), he also holds the distinction of being the only player to ever play in an All-Star Game as a Pilot; Mike Hegan also was selected to the team as a reserve, but did not appear in the game. The following season, Mincher slugged a career-high 27 homers as a member of the Oakland Athletics.

Mincher served as the first president and general manager of the Huntsville Stars, the Double-A affiliate of the Oakland A's (1985–1998) and, later, the Milwaukee Brewers (1999–2014). He served in this role from 1985 until 2001. In 1994, Mincher and a group of local investors purchased the team from Larry Schmittou to keep baseball in Huntsville.

In 2000, Mincher was named Interim President of the Southern League, where the Stars play, when league president Arnold Fielkow left for an executive position with the New Orleans Saints of the National Football League. Mincher resigned from his position with the Stars when his group sold the team to Miles Prentice in early 2001. This cleared the way for the Southern League to remove the interim tag and they made him league president beginning with the 2001 season. He served as league president until retiring in October 2011, at which point the league named him President-Emeritus.

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