A few thoughts:
No matter where I fly, I ALWAYS pack a carry-on with belongings for the first 24 hours just in case the AIRLINE delays my luggage. SO it's no different when I take DME. I really don't care when my luggage shows up, since I'm prepared for the first day, anyway.
Disney says your luggage should get to your room about 3 hours after you arrive at your resort, not 3 hours after your plane lands. It says so right in the DME packet.
By getting your own luggage, you delay yourself getting to your resort anywhere from 15 minutes to an hour. You not only waste time at baggage claim, but you also will likely not be on the bus with others from your flight going to the same set of resorts, so you'll also have to wait for the next bus going to your destination.
Also, flying in with 2 young kids at about 10:30 p.m., I was THRILLED to get to my resort ASAP and get them to bed rather than wasting precious time waiting at baggage claim, etc. Just my opinion ....
If your airline misdirects your luggage, you'd normally have to wait in line at your airline's baggage office, fill out forms, and then get to your resort WAY late. If you put DME tags on your bags and check-in at the DME Welcome Center, DME will act as your agent and track your bags down for you and deliver them to you. I'm sure a substantial number of "late luggage" incidents were caused by the airline sending the luggage to MCI (Kansas City) instead of MCO and then flying it back to MCO, or something along those lines. If you have DME tags on those bags, the irony is you don't have to do anything, and have no way of knowing to be annoyed with the airline and not DME. But if you didn't have the DME tags on your bags, you'd have wasted time at the baggage office in the airport.