I could generate a PGP key for Bill Gates right now, distribute it to the keyservers and sign a message saying that "I owe simias on hacker news one billion dollars". You could verify that the message was signed by my key beyond a doubt but so what?
Bitcoin doesn't have this problem thanks to PoW, when a miner submits a newly mined block the network doesn't need to check the miner's credentials, the block itself proves (statistically) that the miner did a certain amount of work to come up with the block. That's all that matters and the coins/fees are credited to the miner's address. That works because all the data necessary to create the consensus are available "objectively" on the blockchain for all to see. It doesn't matter if the miner pretends to be Abraham Lincoln, that's not what the blockchain works with.
Most problems don't have these properties, if I tell the blockchain "I have a brand new PS4 and I'm going to send it to X in exchange for n bitcoins" the blockchain has no objective way to track this transaction.