Are there any active attempts at finding alternatives to proof-of-work?

The fundamental problem seems to be that, when positioned against the same threat model as Bitcoin, all the PoW alternatives reduce to an obscured form of PoW. Put another way, a PoW alternative proposal must be designed such that participants can't increase profits expending more energy (for example, the fact that stake-grinding is profitable shows that Peercoin's PoS is really PoW-by-obscurity). As far as I know, this is still an open problem.

There's an interesting primer on the subject here: http://www.truthcoin.info/blog/pow-cheapest/

If you can accept centralization (which for many non-crypto-currency use-cases actually makes sense, since it mirrors the way the real world works) - you don't actually need proof of work.

You can still get the benefits of verifiability, append-only, detect misbehavior etc without it. See for example Certificate Transparency (RFC6962) for a standard that implements those properties using similar Merkle Tree constructions.

Disclaimer, I have a startup that generalizes the same Verifiable Logs (and also allows for Verifiable Maps): https://www.continusec.com/

But if you can accept centralization, then why not just use a conventional DBMS? (This is a real question, not a rhetorical argument.)

So that you (or other interested parties) can verify the correct operation of the centralized authority, and so that the central authority can prove that they aren't hiding anything.

For example, in the case of certificate transparency, that the CAs aren't mis-issuing certificates, and further that the logs aren't conspiring to hide entries (such as via split views).

Yes! IMO the most promising approach is Ethereum's Casper protocol. See http://vitalik.ca/files/mauve_paper3.html for more details.

Burstcoin seems to be on the up and up. Proof of storage.