> 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)

Because of Sybil attacks.

> 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.

Remember, the goal isn't to punish Alice. She's losing 100% of the bitcoin no matter what happens. The goal is to assure Bob (or Charles) that the transaction they receive will be recognized as valid by the rest of the world.

If you're Bob, and you know that Alice can invalidate the transaction (e.g., burn the bitcoin you just received) by signing a second transaction in the future, why would you ever accept Bitcoin?

Don't punish Alice. Assure Bob.