No, they wouldn't have to be about him. However, right now they are primarily an attack on Trump, but done in an effective way. That can't hurt the party as a whole.
Second, I suspect that Bloomberg doesn't have a ton of faith that the current menagerie of Democratic candidates have any chance of beating Trump.
You keep writing that the conventional wisdom is wrong, that the Democrats only win when they swing progressive, not to the center. I don't think it's a progressive versus center question.
I think the Democrats win if they can produce a candidate that has charisma, is a good public speaker, and isn't saddled with any major baggage.
My stapler has more charisma than the entire field of Democratic candidates as assembled. Bernie Sanders is the hapless neighbor in pretty much every sitcom ever written. Klobucher is the little girl in the back of the classroom who always knew the answer and always had her hand raised, but somehow was completely invisible to everyone else in the room. Warren is the third grade teacher that every male in the U.S. remembers, fears and hates (to this day) with a passion. These are the people you think are going to get elected POTUS?
No wonder Bloomberg is worried.