Making profits off the team, as you stated, is def a big factor for them. As you said, having an owner who wants to win at all costs, or one who even doesn't mind running the team at a marginal loss each year who can easily recoup those losses when they sell the team for triple what they paid for it, is best.
But Rogers overall bottom line, irrespective of whether the team is making profits or not, can play a big role as well. If their overall profits for the last quarter are bad, then it's harder for them to justify pouring more money in to the baseball team even if the team is breaking even or turning a little profit. Another good example, by Dick_Pole above, is that they recently laid off a bunch of employees. The optics of them laying off a couple of thousand people only to pour 5 million in to the team at the trade deadline looks bad. They may be willing to do that still, but the perception of their corporate culture may be difficult to do.