You're describing an extreme situation. I fly about once a month, airports all over the country, big and small. Never had anything remotely like a 45 minute wait for luggage except at my home airport when arriving late at night and there are few workers so it's slow.
90% of the time you go to the luggage claim and wait up between 0 and 10 minutes before luggage starts coming out. By the time you tell Disney what you want to do and then they try to convey your request to another company its likely spinning on a belt....alone. It has no yellow tag. Its just like any other bag for that airline. It will end up on the belt at bag claim, the only spot where anyone including yourself will get be able to get it (either be it DME, or your airline, or a thief). Telling Disney to get a bag with no yellow tag means they can't get it at the normal time they grab yellow tags. They would have to go on a hunt for it.....probably starting with the carousel. Since you didn't tag the luggage at all, they may even tell you to just go and get it (since you know what it looks like) and bring it to them.
Why? It would be much quicker in 99% of all situations to just grab your bags than to ignor them, wait in line, and tell another company to tell the first company to go and pull them off the belt that you chose not to stand next to.
This isn't about "trusting" DME to do something, its about doing the easiest and quickest thing to get your bags where they need to be......with you. You're asking DME to work with a bag that's not been yellow tagged, it's completely out of their system. If DME accepts your mission request, I'm sure DME will try very hard to figure out how to find your bags (that you choose to leave on the carousel) and the airline may go and look for it based on descriptions or numbers and eventually (assuming your luggage isn't stolen) they find it, pull if off the carousel, and give it to DME). Or you could just grab it and go....either way works, one is just much more likely to work.