Totally agree that there should be some mechanism for preventing keys cluttering the inventory. I can see benefits of all three suggestions (separate tab, key ring, keys vanishing once the lock(s) they open are unlocked).
Because there’s a better than even chance that I’ve lockpicked the lock before finding the key, I’m not sure vanish on actual use would help. Though in some cases I possibly do have keys but automatically lockpick anyway because the chest is locked. I don’t think there’s any difference between the tooltip or right-click menu between a locked chest/door you do have a key for and one you don’t?
I think I have a slight preference for keys vanishing once there are no more locks for them to unlock. If it were more feasible to keep track of which keys went in which locks then I’d say keep them, so even if we didn’t need them that playthrough we could see there might be an option for another one. But as it isn’t feasible, keys vanishing when they’re of no more use would be an incredibly useful indication that there’s a locked chest or door I’ve not yet found.