Probably this is job politics more than anything to do with prudence over boosters. I left my last job in part due to micromanagement from offsite idiot professors, same concept but in this case the idiot is an idiot to end all idiots. I likely would have done the same thing in their shoes.https://www.google.com/amp/s/www.newsweek.com/disagreement-wh-over-booster-shots-prompt-senior-fda-directors-step-down-1624750?amp=1
Two directors from CDC step down after booster shot disagreement with WH
On the flip side of that, both the FDA and CDC are notoriously slow in approvals. These agencies, like so many other government agencies, have an absolute horror of ever recommending or approving something that turns out to be wrong. They apparently see being wrong through inaction as the lesser evil so they are conservative to a fault when it comes to approvals. This is not to say that this behavior is necessarily incorrect given their position, I wouldn't know about that - it is more a matter of illustrating why they are resigning over the WH bypassing them.
They have a point I think, based on what is known about how the vaccines perform over the course of several months and how little we know of longer term effects. The WH is pursuing happy talk policies that are really quite unscientific because that is what the media and the public focus on. A great deal of the confusion over these vaccines comes from that habit of politicians and media on both sides of the big idiot rodeo that we call our political system.