It is the divers' responsibility to make sure his/her gear is set up properly. You are correct. This is why I set my own gear up. I can't understand why people want someone else to do that for them. Having another person messing with it just adds to the possibility of error IMO. Yeah, yeah, the DMs do it every day, but they don't know the individual piece of gear and the divers' specific setup. Only the diver who uses it does. If you can't set up your own gear you shouldn't be diving.
...and setting up gear is not a DM's basic function. It's something that was added to try to get extra tips. Next some lazy diver will want the DM to dress them in their wetsuit.
Cozumel is much different from most Caribbean dive destinations. There are many dive ops to choose from and you can find virtually every combination of service and price to suit just about any divers preferences and taste. There's everything from just a boat ride, bottle of water and a DM, to full-service "valet diving," and everything in between.
Choosing a dive operator who will rinse and store my gear and have it set up on the boat for me when they pick me up in the morning is my choice. I'm very capable of doing all that myself (and bringing my own towel, snacks, drinks, etc.) and do on most of my trips to other destinations. In Cozumel I have a dive op I trust and who I pay for the extra service and amenities because it lets me enjoy my vacation a little more. I still check my gear before every dive and do a buddy check. I also analyze my nitrox tanks.
Even with the premium dive ops, if you prefer to set up your own tank and rinse and deal with your gear yourself everyday, they won't object. You won't get a discount for doing it all yourself, but they will let you do it.