I hope we're not talking at past each other, but I read your post as proposing a system where there's a central party that computes transactions "in the open" in a way that third party observers can verify.
That's hardly useless, but it's not a replacement for Proof of Work. PoW is for decentralizing the ability to choose between competing valid blockchains. It prevents double-spending by making the benefit of double-spending (the value of the your transaction) far less than the cost of double-spending (the electrical cost of 51% of the mining power for 1 hour or so).
Your system uses a centralized host (see "Services with Secure Ledgers", paragraph 2), and (I presume) third-party observers can verify a "secure ledger" by seeing which one has been more recently signed and time-stamped by the single host.
I mean, sure, centrally hosted servers are more efficient than Proof of Work. No one who know what they're talking about disputes that. But the whole point of PoW is to allow decentralization without a single host.