I'm sure it would. When and how is going to be a different question, and if that is an acceptable risk on your part is another question. Where this lies on an ethical scale is another thing. There is liability as well - are you expecting the carrier to take responsibility for the bag if it is circling the carousel while you decline to pick it up? I wouldn't.

Also, whether they'd pay to deliver it to you is yet another question - I'd guess if you just left it there, they would require you to go back and pick it up at FRA, after all, why should a carrier be responsible for paying to ship your bag to your house/hotel when you purposely left it there (as opposed to if the carrier misplaced it). They are already spending resources because you decided not to follow procedure - why should they be expected to pay a delivery tab, too?

I can tell you about an experience involving Canada on the topic - I once had EWR-YYZ-YVR on AC metal the whole way - EWR-YYZ was pretty late (maybe 2 hours - AC fault) and I was going to likely misconnect. At baggage claim at YYZ, I basically asked what I could do - they had me fill out a delayed baggage and customs form, and called the gate to let them know I was coming. Got through customs and ran through security (obviously, not literally) and arrived at the gate right about departure time, maybe a minute or so prior. Aircraft was gone. GA was still there and basically told me they don't hold NRT flights for anyone (since the flight was going onward from YVR - forget about the fact the ground time would have been a couple of minutes less then the scheduled 2 hours). It just shows you that you lose some. My bag was delivered he next day, but obviously, the delay was ACs fault so they delivered. I wouldn't expect the same if you just left it there of your own valition and didn't bother filling out a form.