Yeaaaah, most (all?) helicopters with conventional tail rotors mounted atop vertical fins send the driveshaft up the leading edge. What's missing from the 139 is not the driveshaft but the push-pull pitch change tube, which probably left with the gearbox.
Just one word on the "birdstrike" scenario: Maybe. Let us accept that the bird did not overtake them and fly into the tail rotor from the rear. Unless the bird dove vertically from above, it's hard for one to get the tail rotor without going by the cockpit pretty close - and there were two guys up front presumably with their eyes open who ought to have seen it. We all take work hard to avoid birds (the ones we see, anyway). And maybe they did.
Right now all we know for sure is that "something" caused that tail rotor gearbox to depart the aircraft in a big hurry. As with any accident, we'll surely find out more in the days and weeks to come.