You can prove a doublespend instantly by showing two conflicting transactions both signed by thar party. This pair can be distributed as a proof of malice globally in seconds via a push messaging mechanism.

There have been lots of e-cash schemes proposed in the academic literature that work like this, or variants of it. Schemes where participants are anonymous until they double spend are popular.

Let's re-write your proposal but substituting the word notary for miner:

To profit, the miner would have to be sure the payout from agreeing on collusion (or to perform the doublespend themselves) would pay out better than acting honestly for a given amount of time info the future. This means transactions for small sums are secure.

That's the exact argument we're having. The assertion is that a "rational" notary would kill his own business to increase his profits in the next few hours. So you're just arguing that a notary is different to a miner, without spelling out exactly why.

Does the notary have to make a big up front investment? If so, why is that different to mining investment?

Is the notary non-anonymous and afraid of being charged with payment fraud? If so, note that big miners do lots of non-anonymous things too, like renting warehouses and importing specialised equipment.

Is it because of the big up front collateral they're meant to have lying around? If so, how do you ensure a fluid market for notaries?