The 'you own the keys' argument doesn't really hold up imo, after all, you 'own' your creditcard too, still the company manages it and in the end has ultimate control over it.
Come back with an argument as to why this app is broken when you understand how the private keys in Bitcoin work.
There is a certain issue with trusting the balances coming in (if for any reason the server providing the data is motivated to lie to you and tell you a fictitious transaction confirmed when it has not, there's no way for you to tell that happened without the blockchain), but beyond that there's very little harm that can be done with such a "centralized" solution.
Again, the great thing about Bitcoin is not that it has to be decentralized, but simply that it can be. You can be your own bank, your own currency exchanger, etc. Or not, if you simply don't give a shit about those things.