Isn't PoW the defining characteristic of a blockchain?

Just to give a bit more detail with a blockchain you need some way to limit who can create a new block since you can change history by building a chain on an earlier block if you can create as many blocks as you want. In public networks this is hard since people can create as MD Danny accounts as they want so PoW which limits who can submit by resource usage or something similar is needed. In a private network you know all the participants so you can achieve the same result with something as simple as round robin.

To me that feels like just a general consensus database problem, which has been solved a long time before bitcoin came along and "blockchain" gained mindshare. It appears "private blockchain" is mostly trying to ride the buzzword hype if it doesn't incorporate the new ideas and solutions that Satoshi presented, which is mostly about generating consensus for a ledger among a set of anonymous and possibly hostile peer-to-peer nodes on a public network?

Right so it's just a different way to solve the problem.

Satoshi didn't coin the term blockchain and they existed for a long time before bitcoin so giving him sole claim to it because it paired it with a feature that isn't specifically required to have a blockchain seems silly.

That said I do agree that their are a lot of buzzword vultures using it at the moment for stupid business ideas though I think 99% of the bitcoin business ideas are poorly thought out and poor replacements for existing solutions("but with bitcoin!") so it's no surprise 99% of the blockchain ones are too.

No, it is just a good(the best?) way to do public blockchains.