That early in June, I am pretty sure the switchbacks won't have a beaten path through them, so you will need to ascend the chute.
I suspect the bad accident was due to people wearing microspikes and not crampons, but I do not know that for sure. One thing I do know is that the accident happened early enough in the morning that the temperatures were cold and the snow surface was frozen and icy. There is NO advantage to arriving at Trail Crest before 9 AM (10 AM is even better!) -- that gives the snow surface time to warm in the morning sun, and you will have much better traction.
On May 1 at 7 AM, all the won slots that have not been paid for will become available to the fastest fingers, so you might try to get a slot for your group later in the summer. If you do, be SURE to release your reservations, so others can get them online.