My perspective: There was zero to lose by trying the gambit. I mean, goading Donald Trump generally results in a lawsuit at worst, and suing the president for slander is exactly the sort of thing that would make Trump effectively go away forever. There was plenty to gain: if you can get him to charge, it will be entirely on the Republicans to stop the stampede. So the costs-benefits analysis is this: do you let the press writers to write a few throwaway jabs about Trump and birtherism? Or do you make a few phone calls, say "hey, I've got an idea" and then get those same writers to really get under his skin?
The only remaining question is whether you think Hillary Clinton is playing that many moves ahead. I would argue that you're stupid if you think she isn't. I'm not betting that this was a known outcome that she was banking on; I'm betting that this is a possibility that she's been cultivating into a probability.