There is inherent problem with digital assets, namely they can be perfectly copied. Until Bitcoin came along, there was no decentralized solution to digital scarcity. If you wanted a digital asset to be unique, you had to rely on a central authority who kept track of who owned the "real" one. Think of a piece of software that you were asked to register with the company. That's how digital scarcity used to work.
Bitcoin solves the problem by creating a single, public ledger that is copied between nodes on the network. A protocol was defined for the nodes to use in order to come to consensus on the state of that ledger. This protocol has some ingenious protections against malicious use. Specifically, the proof of work component makes it fiscally unviable to "cheat" for a sustained period of time.
Encoding an owner into the digital asset itself, does not solve this problem. Let's pretend a bitcoin was like a file that you can store on a hard drive (it's not, btw). Let's say Bob has one that has encoded in it some ownership indicator that says "I belong to Bob". Let's even suppose that this indicator was cryptographically signed by whomever gave the bitcoin to Bob. What stops Bob from making 10 million copies of his one bitcoin? If he did so, it would be cryptographically signed by the person who gave it to him, and Bob would have the ability to sign each one over to someone else.
In the case of my "system", you would prevent copying by generating a
unique random seed based off the keys of the two agreeing parties.
Kind of like normal serial numbers in paper bills now, ensuring each
two-party coin transfer agreement has a unique id, hence cannot be
You are incorrect; it could still be copied. The end result is still copyable, and it would do no good without an authority to verify the unique number with. If Alice gave Bob one bitcoin, digitally signing it with a unique random number agreed by both parties, and then Bob made millions of copies, who would ever know? Bob can pay Charlie and Dave with copies of the same coin, and neither would be any the wiser. This would require a central authority that knows about the Alice->Bob transaction, as well as the Bob->Charlie and Bob->Dave transactions. Recipients of payments would have to verify with this authority in order to be able to trust Bob. This solution solves nothing.