The smartest thing Cozumel could do is make that arrival process more comfortable and less stressful. Here are some ideas I have off the top of my head:
1) Kicking out those Timeshare salesmen.
2) Have all the rental car vendor counters there
3) Have visible pay phones
4) Normal taxi service.
5) An information counter staffed with multi-lingual people who are not out to sell you anything.
6) A bank our money exchange counter with a fair exchange rate.
Then we can relax, have fun and enjoy what another culture has to offer. Please don't tell others to "Just Relax" unless you truly understand the stresses they are going through. That airport is just not friendly to arriving passengers.
I probably should have read the rest of the thread before responding, but...
First of all, this is Cozumel, not Baghdad. Or Houston, for that matter. Your loved ones were safer on the curb there than they would have been in most major US cities.
Second, your stress was brought on by your wanting things to be other than they are. Get over it and go with the flow.
1) The timeshare folks are a minor irritation at worst.
2) The best way to deal with a rent car is to get yourself, your luggage, your family, etc. to your hotel first, then go get the car. Or not, if you are staying close to or in town. If you try to deal with getting yourself around by driving in a place you have never been, you are just going to add to your stress level. Cabs are easy and cheap on Cozumel, and it's a small island.
3) If you take the shuttle to your hotel, you don't need a phone.
4) The taxi service is normal, but it's different. Taxis taking folks from the airport would make for a terrible traffic snarl. Take the shuttle to your hotel.
5) The shuttle window folks are fairly fluent in English, though all they really need to understand is the name of your hotel. Tell them where you are staying and they will tell you how much the shuttle costs to get there in pesos or dollars.
6) You don't need a money exchange in the airport; you can pay the shuttle fee in dollars and worry about getting pesos later.
The problem as I see it is that you expected the airport on Cozumel to be like a typical cosmopolitan US airport, which is an airport, a rental car terminal, a bank, a restaurant, a communications center, a social services center, a shopping center... The Cozumel airport isn't all that; it's just an airport. All it is concerned with is getting folks on and off planes; if you have other needs, you need to handle them elsewhere.
You really need to deal with things as they are rather than carp about the way you wish they were. THAT is where your stress was coming from, IMO. But you've done the hard part now; now you know how it works and next time you'll be fine.