We just disembarked in Manhattan on Thurs. Port staff were directing passengers across the street for taxis and rideshare, but if you tell them you have private transport you just go upstairs in the terminal and get picked up there. You are outside for a moment waiting for the elevator but it's under cover.
We ordered an uber, but instead of saying "uber" or "rideshare" we said the magic words "private transport" and were allowed upstairs. The uber came in 3 minutes. No one stops them from driving up to the terminal. If you used a car service you'd also pick it up there upstairs.
Also, if he wears the boot off, the staff may be more flexible. But just remember the magic words are "private transport" and you'll be ok.