I don't know any coins that operate without either Proof-of-work or Proof-of-Stake. To make your ideal a reality someone would have to invent a new way to power a block chain, unless I missed a coin that quietly implemented a new system without many people noticing.
There is a way to power a blockchain without proof of work or proof of stake, and we're doing it on the Carboncoin
It's not the perfect solution, because as the op is well aware, a perfect solution does not exist
We have demonstrated our ethos and trustworthiness by operating without security breach for over 2 year. This becomes more valuable the longer it goes on.
Anyone can hash on the network but they wont be rewarded with enough coins to make it worthwhile to do that in anything other than the most effective way.
I have my "ideal" structure for a coin, which will be implemented on the carboncoin project at some stage in the future but in the meantime we have been able to add approximately 500 non crypto users since we started (who have bought coins coming from the market), and that is without any significant fundraising or resources for marketing or development.
The funding is now happening, so we will see where it goes. I would very much like to engage this community and find fellow innovators to work with on the ideal currency application of the blockchain - Carboncoin continues to be my best effort