No US phone number: how will the captain reach me?

Give the desk the number you will actually be reachable on, whatever country it belongs to, and say how you would rather be contacted. Most people landing from abroad have a phone that will find wifi in the terminal long before it finds a signal, which is worth saying when you book rather than discovering at the kerb.

What makes this work is that nothing depends on a phone call in the first place. The captain is assigned to your flight, the pickup moves with the aircraft, and the meeting point is agreed before you take off. The call is a courtesy, not the mechanism.

If you would rather be met inside arrivals, ask for a meet and greet and the question answers itself: someone is standing there with a sign, and nobody has to reach anybody.

You will also have the ride's own page — the link sent when the booking is confirmed — which shows the captain, the car and the plate, and where they are once the trip is live. It opens in a browser. There is no app and nothing to install on an aeroplane.