Fortunately such "proof of burn" tokens already exist for a while, so we can analyze how their price evolved. To be fair, none of these had the intention to be pegged to a fiat currency, but the mechanics are the same.
One of the first tokens of this kind was Counterparty (XCP) on the Bitcoin network. It was issued in 2014, with XCP tokens being distributed according to the Bitcoins burnt.
However, this made not behave XCP in a "stable" way, not even compared to Bitcoin:
Source: Coingecko(orange line is the price of XCP in
BTC, blue line in USD).
You see that even if you can argue that XCP always was "backed" by the "collateral" of the BTC which were burnt when they were created, the price lowered year after year, with some spikes in the middle.
The reason is actually:
a currency's value is not only determined by the money supply, but also by the demand. If XCP, or your "USD pegged" token, had a perfect stable demand per token then it could hold its value against the USD. But this is so highly unlikely that it becomes totally unrealistic. Demand varies for various reasons, from seasonal changes to economic cycles, and of course cryptocurrency market's own attention cycles.
So you always need a burn mechanism for those times when the demand goes down. And that's what algorithmic stablecoins like Dai actually provide, and that's the reason why they can work.