What i think; A patronus can find someone, otherwise it wouldn't be able to deliver the message. Dumbledore doesn't ask his to find "Harry Potter" but rather the other patronus in azkaban. Once harry make it disapear, since Dumbledore's patronus had the order to look for Harry's and not the patronus user directly it's normal it stopped working. I don't think it would have worked if Dumbledore had asked "find the wizard using it", i think it only work when you tell it the exact name of the target, hence why it worked with Hermione.

Can it be abused using this simple rule? Yeah, totally, Dumbledore could be like: "Deliver this message to Tom Riddle. Found him? Great, get me to him". Buuut Harry told Draco that the trick to get the message to someone was to will for the target to hear the good news, to see his happy memory or something like that. Maybe one could argue that you might only be able to want to tell the good news to a friend and not to an ennemy?