What I do not understand is why we couldn't build this onto a web of trust + proof of stake. It may require some more state during resolution, and might even require falling back to automatic service denial if there is a severe network interruption (say in the event of nuclear war), but I can't figure out why we couldn't have miners just sign a growing dag of transactions and once a subset of the dag is accepted by 51% of the stake that network can prune the excess data (all the signatures). Work timestamps into the signature process and pin the keys used to sign the dag as only those belonging outside the current resolution. If a key is used to sign a transaction that conflicts with it's own "Verified & Locked" signature then it's coins are burned, including for it's own transaction. So in your example, if Alice signs both transactions and she doesn't have the funds to fulfil both of them the transaction causes Alice to lose 100% of her money.

I'm sure there is a reason, and I vaguely recall some people talking about proof of stake a couple years ago, but I'm surprised we're still melting icecaps running ASICs out of China for new coins.