I think the main way Professional Democrats are bad at messaging is that they are generally unwilling or unable to take the lead, so everything ends up being reactive. Conservatives talking immigration? We'll respond to that. Inflation? We'll respond to that.
When there's an obvious opening - some conservative steps on a giant rake somewhere - they come out for a day and then hand the ball to the press thinking they're supposed to run with it. Whether they should or shouldn't run with it is a question, but they won't do it on their own.
The way something "bubbles" up is so that journalists won't shut up about anything else, temporarily, is complicated, and changes over time. Once upon a time Matt Drudge Ruled Their World and it was as simple as Drudge headline at 2, on CNN by 4.
That's no longer the case, but the decline of Drudge left a big hole and it was filled by... the rest of the grifter wingnutosphere. Lost opportunity.
Anyway, there was a moment, as I said, when a different path was possible. Oh well.