Is a solution in which control of the network is secured through the same private keys that secure the stores of value an option?
No, you've hit the nail on the head. In fact it is a generally abstractable property that proof of stake reduces to proof of work under adversarial conditions as those with stake grind out possible recent block histories that lead to their own stakes being selected to select the next blocks.
If it's worth saying, but not worth its own post, then it goes here.
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