Or what would happen if improved cryptography comes along, how would the protocol deal with this? Would a layer on top of the current protocol be enough to patch it up or will Bitcoin die?
It could very well already be broken.
It would allow the organization that has the backdoor to potentially double spend, reverse transactions, issue coins and so on, depending on how wide the backdoor is.
If people would catch on to it then Bitcoin will be hardforked and everything will resume as if nothing happened.