It might be an unpopular opinion but I think it's more plausible than the notion that a team of professional players is mentally so devastated that their beloved 1.5 half season long coach goes to a rival that their performances drop multiple levels, especially if that was already the case two or three weeks Rose told them he'd go at the end of the season.
To be honest, I think it's almost a bit cute to think that Rose's departure has more influence on the results than the fact that the team not only lacks depth but also quality while playing three competitions during a tightly scheduled Corona season. But that's just me.
I'm sorry, I always found the idea that professional football players are so confused by a few media reports that they can't focus on their performances a bit silly but at least you can argue this can mess with their focus and nerves on the pitch. But a coach? How should this even influence him in his work? As if it somehow tampers with his tactical approach, training routines, analyses and so forth.
Football at that level deals in fine margins. A couple of weeks ago things were looking dire for Dortmund and a few good results later the confidence is back and a top 4 finish and cup success seem at least possible. For Gladbach this is especially true, since they don't have life insurance players like Haaland or Sancho to bail them out. They need teamwork, workrate and confidence to succeed and in that regard it makes a huge difference if the guy who is supposed to motivate and lead you has already secured his next step up, perhaps broken some promises or dashing some hopes as collateral damage. It's probably a difficult situation to manage a season that had such a hopeful beginning slowly petering out, with some key players probably considering their future as well.
Management and players at Gladbach can read the news, too. They will have had their questions for Rose. It's something he probably had to do a lot of dancing around.