Because when the public hears about other reasonable people showing hesitancy, it shows that the media narrative isn't as true as it is made to be.

I imagine it is to the advantage of both pilots and SW to not admit it is about the vaccine, because the media would take advantage of that fact otherwise.

Also the pilots can't admit to a sick-out one way or another to avoid legal repercussions.

Not saying this issue is because of the vaccine mandate, but it would make sense with the way this story is playing out.