Sounds nice, but may pose some challenges.
HashCash, bit gold, and b-money all tried to use self-mined proof of work as cash directly. Those projects were unable to solve the doublespending problem as they did not feature a global consensus system. DigiCash on the other hand was able to solve doublespending but required a central operator to clear payments and issue new tokens—shutting down the operator meant that the currency stopped existing.
Bitcoin's great breakthrough was that it provided a practical solution to the doublespending problem without a single point of failure. It combines the strengths of the above two approaches into a system where network participants both issue coins and decentrally author the global state. The latter removed the need for a central operator, while still providing the referee function which enabled every participant to locally resolve the doublespending problem.
What you describe as
"some kind of (self-mined?) token that can be proven to be unique and to be "owned" only by the holder"
is essentially a description of the doublespending problem: how do you guarantee that a digital good cannot be transferred multiple times?
So, your proposal boils down to a suggestion that we would "only need to find another, even better way of solving the doublespending problem" and then we'd be able to create an even better cryptocurrency. Sounds about right, but might be a bit optimistic since this solution already took a few decades to emerge, and the requirements you describe sound a lot more challenging.