You are absolutely right that it all begins with ownership. Look at Billy Beane, there's a reason he's stayed in Oakland. Ownership lets him do what he wants. He was even given the luxury of no less than 5 consecutive .500 or less seasons right in the middle of his ten year tenure. You only get that kind of rope from an organization that cares as much about cost cutting as about the product on the field. Beane's no idiot, he realizes that there's a kind of freedom that comes with not having the pressure to win all the time and he ultimately turned down the Red Sox job because all he really he wanted from them was recognition not the actual situation. Trying to stretch every dollar in Oakland is a challenge he relishes and he's good at it. He works well with the limitations put on him and the trade off for those limitations is more than off set by the fact that he's given more freedom to make mistakes than probably any other GM. And it's not at all crazy to think he works better with a limited payroll than he would with a large one. Limitations stimulate creativity (necessity is the mother of invention and all that).
As for Friedman, we really know much less about him than we know about Beane. As has been pointed out, he's a finance guy and of a for all we know, his long term ambition may have nothing to do with sports or he may indeed want to stick with baseball but have a career that's more in the Pat Gillick mould where you preserve your legacy by bailing at the first sign of the downturn. Things have definitely gotten more difficult for the Rays since baseball has eliminated the compensation system that they gamed so well. Things like picking up roster players in the Price trade could help to prop up the window for a little while longer but it will be really hard to keep it going without the kind of MiLB depth that they used to have. The challenge for the new GM will be to refine the approach, find new loopholes if there are any, do more with less picks and try to avoid a drought period as adjustments are made.