Make sure with the satelite phone you know who to call. The Park Service may not be quick to respond, if at all, to a flat tire but maybe faster if you are bleeding. Calling a family member may give THEM some peace of mind, but may not help you much. So even with a phone, help may be very slow to arrive. Be well prepared to wait a few days for help. Better yet, be prepared to help yourself (fix a flat, don't run out of gas, don't get stuck, [plenty of food and water).
A towing compa may be relatively fast to respond but at what cost? Help may come from the service station at Furnace Creek or from Beatty, Nv., well over 50 miles one way. I'm not sure your road side service (like AAA) will pay for this, so you may want to check first. I also understand all bets are off with rental cars, maybe no insurance at all.