But to decrease money supply and scarcity there is also reduction in volume and trade, exchanges need volume and trade in order to keep Coins listed so if you are losing money on every trade then you won't trade, exchanges have less volume, less distribution of the money supply, less nodes and consensus. IMO Proof of Burn is not as popular as higher inflation because in higher inflation Coins people will sell excess coins for profit and retain most of their Coins once they reach ROI so you get wider and wider distribution over time.
You're assuming that price remains fixed. A reduction in supply tends to increase price - take a look at 42, for instance: 42 total supply, coin value: 3.4 BTC coin.
https://www.cryptsy.com/markets/view/42_BTCInflation reduces the value of coins overall, because it increases supply... In essence what assume is the case is completely backwards from reality

There are no functional proof of burn coins that I'm aware of, which is why I'm interested in exploring how it would work.