Interesting, thanks for the insight. So it seems the main differences between the two, is that Shadow is on a bitcoin blockchain, which gives it some advantages. For example it can plug in better with existing infrastructure, and have benefits of both transparent chain and anonymous chain. Also since only about 1% of transactions are used for anonymity, it will have less blockchain bloat than Monero where 100% of the transactions are done using ring sigs.
If only 1% of transactions are used for anonymity you will have massive timing anonymity leaks and such. If you take transparent coins, convert to anon, pay someone, and he converts back (both of which are logical if most commerce is taking place non-anonymously) then the whole thing is quite obviously traceable.
I don't really see much advantage to a coin that is 99% transparent, and definitely see disadvantages. Just use Bitcoin, or if you really want PoS, use whatever is the leading PoS coin, probably BitShares I guess. BitShares has stealth addresses by the way.
I wonder if this destruction and reminting gives it any privacy advantages.
It does not. The creation and destruction process is entirely transparent.