Can a publicly visible but privately mined block chain be trusted?
I'm hoping that data transparency and data integrity can be preserved by having a publicly visible blockchain but since there's no reward to users for mining there's not much reason they will actually mine.


There's really no point. There's much simpler ways to provide the same data transparency and integrity guarantees. The only reason to use a blockchain would be so you could say "it's cool because we use a blockchain".

You can use transactions without a blockchain. You can map ownership to cryptographic identities without a blockchain. The sole miner just sign that it's accepted a transaction rather than doing anything like mining.

A blockchain doesn't do anything for data transparency. Just because you have a blockchain doesn't compel you to make it publicly accessible. And you can make data publicly accessible regardless of its format, blockchain or not.

And there's lots of costs associated with blockchains, none of which give any benefit. The whole point of mining is to allow you to determine what is truth without a central authority to say what is true. If you have that anyway, what's the point?

This is such an easy to solve problem and blockchains have such complex tradeoffs, it would really only make sense if you were too lazy to build something more appropriate and just happened to have a blockchain solution you could easily drop in.