The mechanism of DAI is quite a bit different. If the value of the underlying collateral crashes the created DAI is destroyed, which raises the value of DAI from its peg [1]. The only scenario in which DAI depegs in a direction against DAI holders is if the underlying collateral (i.e. Ethereum) flash crashes before anyone can liquidate the minted DAI.
Sometimes when I read these things I feel like I'm going crazy, like it sounds kinda logical at first glance but when you think about it it makes no sense. Why would destroying it automatically raise value? If demand stays and supply goes down, sure, but it's not like it's something like food or housing, which people need - why would the demand stay the same?
The demand for DAI comes people who want to repay their DAI denomonated debt to unlock their collateral or stave off liquidation, or from liquidators who need DAI to participate in a liquidation auction that happens if a vault holder's collateral falls below a target threshold. In any of those cases, when the DAI is repaid, it gets burned. As long as folks want their collateral back, or liquidators want to buy that collateral, there is demand for DAI.
[1] https://makerdao.com/en/whitepaper#maker-protocol-auctions