Does it need to be profitable? It seems to me that if it siphons money off from competitors, that's already enough to hurt the competitors relative to itself. In fact not only does the advertising not need to be profitable, even the sales of X do not need to be profitable, for the competitors to be harmed... dumping would be the extreme example of that.

You are talking about a different aspect of the scenario, where the company is reaping profits from advertising. Sure, that's kind of beside the point though. Which is why it's important to recognize that the harm happens even without the advertising being profitable.