I mean it's centuries old and then he's just gonna go *snap* and throw it into a ravine? It just rubs me the wrong way. Sort of like the way frodo left middle earth after saving the world if you know what I mean.
Its probably just so they have a record, and can more easily identify the wand if anything happens. It also, clearly, gave them information beyond weight.
Harry didn’t just use the cloak when he went to the Ministry to steal the locket, so there is obviously other security. I’m sure there are plenty of enchantments on the place that can detect invisible people.