Even that part is unnecessary. In Bitcoin, we have UTXOs, which can hold zero satoshis. So, if that kind of system would just count any UTXOs, and would be deployed on top of Bitcoin, then people could artificially create a lot of outputs with zero satoshis, just to increase their chances of being selected.

But, there are bigger problems than just that: if everything is based only on signatures, and all coins are initially in the hands of the creator, then the chain of signatures can be always overwritten by the creator: in all Proof-of-Stake-like protocols, if you have 51% of coins, then you can rule the network forever.

In case of Proof of Work, having 51% here and now, means that you can only rule the network here and now. And that you have to keep spending your resources, as long as you want to keep the majority. But once someone is beaten by another miner, someone having 51% hashrate can lose its leading position. But in signature-based protocols, like Proof of Stake, signatures can be made without much effort, which means "once 51%, forever 51%" instead.

And in that case, if the network is going to be centralized anyway, and controlled by whoever will hold the premine, then why don't start with just Signet, and explicitly use something, which is based on Proof of Authority anyway?