Kyle Boddy
The next name on this list does not work for a Major League Baseball team, but you may have heard of the baseball company he co-founded called Driveline.
Driveline has become prominent in today's game for its work with Major Leaguers and Minor Leagues to unlock their full potential and help them become the kind of impact players they want to become. Various Cardinals players have spent time with Driveline over the last few years, so there is already familiarity there from different players in the organization.
Boddy was the Director of Pitching for the Reds prior to founding Driveline, and helped vault the Reds pitching staff from the bottom five in baseball all the way up into the top 10, and drastically improved the Stuff+ rankings of their pitchers.
Player development is an area of the organization that has come under fire in recent years, so going out and getting one of the leading names in the industry would be an interesting move. Boddy has interest in joining a Major League front office again, and my guess is that he'd jump at the chance to run the Cardinals.