We'll see how well it maintains its peg if/when ETH and BTC go down to nothing.

Either way, as far as I understand it, the protocol keeping DAI stable requires participants to be able to tell how much money has been deposited by a particular wallet (so that the agreed upon collateral requirements can be enforced). If that understanding is correct, then DAI requires pseudonimity instead of anonimity, and so can't have any of Monero's strict privacy guarantees.