Klain got lucky too. The elections went a lot better than expected for the dems and so the need to move him out softened a bit- although he is a key part of team Biden and despite various successes dems might like to cite, Biden's poll numbers and support amongst voters, including dems, is El Sucko. And even those who think he did okay nevertheless do not want him to run again, for the most part.
The press secretary thing is a disaster though and he needs better messaging there. You only use and abuse Admiral Kirby there so much to compensate for her. They made a good move a while back to get her out of Ukraine and international affairs press secretary work by bringing Kirby closer in but that its limits. One day I was sort of blown away to see him answering press questions on inflation and the economy. What?
I hear ya on the Vilsack, Landreau, or Sullivan thing but you may be thinking this through rationally, whereas Biden starts with the skin color or gay/trans "qualifications" and works backward from there. Some straight whitey-tighties in the ones you mentioned.