You can not and should not get a reclaim check until you hand over your bags.

Otherwise you will just board a flight never checking a bag, and when you get to your destination claim your luggage was lost, you handed it in, see you have a claim check, pay me money.

Claim checks can only be issued when the bag is turned over.

Boarding passes are completly different than luggage tags when discussing printing at the airport.

Every bag needs a tag, and every bag tag will needed to be printed at the airport. There's no real way around it, they have to be standardized and having somebody print a tag on an ink jet printer and leaving the bag out in the rain causing the ink to all run is not ever going to happen. No amount of mobile use will ever change the fact that for the foreseeable future, bag tags need to be printed on a printer at the airport. As that has to happen, printing a claim check at home will do nothing to change the volume that has to be printed at the airport. Won't change the people needed, won't change the counter space needed.

Boarding passes are totally different. You don't need to attach a boarding pass to a piece of luggage when it's handed over. Allowing people to print boarding passes at home or at a kiosk has considerably cut down on the staff needed at the airport, and in many cases the lines at the airport.

If you have trouble holding on to your tags, have you considered sticking them to the back of your phone?