This is just about the pitching side of things, Rothschild was brought in by Girardi and it was time to update things. Lindsay Adler in The Athletic has details about how the shakeup is organizational:
"Although Rothschild is the only announced change to the major league pitching staff, his dismissal comes as the organization is turning over nearly their entire pitching development infrastructure. In a radical overhauling of their minor league coaching system, the Yankees fired the pitching coaches at nearly all of their affiliate levels.
The changes throughout the organization have been taking place over the course of the season. Danny Borrell, who spent 11 seasons working in the Yankees minor league system and the last five as their lead pitching coordinator, left to become the pitching coach at Georgia Tech. The Yankees then hired Sam Briend as their new director of pitching, a role that would have overseen Borrell. He was previously the director of player development at Driveline Baseball.
The Yankees also recently dismissed Scott Aldred, who was the pitching coordinator at the high-minors level. Other changes include: Tim Norton, pitching coach for the Double A Trenton Thunder; Gabe Luckert, pitching coach for the Low A Charleston RiverDogs; and Justin Pope, pitching coach for the Gulf Coast Yankees.
Tommy Phelps, the pitching coach for the Triple A RailRiders, was retained.
It should be expected that the Yankees finally turn fully “new school” in their hiring process for a new pitching coach. Their analytics department is one of the most extensive in the major leagues and they have invested heavily in the new pitching technology that assists pitchers in pitch design and in refining their mechanics. Briend will likely have significant influence as the Yankees go through the process of hiring a new major league pitching coach.
Some teams in the major leagues are now carrying two primary pitching coaches on their staffs now. Across town, the Mets replaced pitching coach Dave Eiland in June with 82-year-old Phil Regan and 37-year-old Jeremy Accardo as a pitching strategist. The Braves employ pitching coach Rick Kranitz and bullpen coach Marty Reed. The Yankees have done well with the double-barreled approach from hitting coach Marcus Thames and assistant hitting coach P.J. Pilittere."
https://theathletic.com/1331317/2019/10/28/larry-rothschilds-firing-highlights-a-radical-overhaul-of-the-yankees-pitching-development-infrastructure/