Well that's not good. Regardless if the diaphragm looks OK or not, something is leaking there and it either needs to be addressed or replaced. Sucks that something like that would happen with a new pump. Is it a new-new pump, or a new-old stock pump? In any event, a ruptured or leaking diaphragm there could account for both issues. The leak and the lack of function.
I guess it's conceivable that a stuck closed output side check valve could allow the pump to suck in a charge from the inlet side, but then force that charge out of a hard to see, small hole in the diaphragm.
And, yes, change the oil. You don't know how much went down there.