PoW makes sense from a first principles approach [1]. I don't see Handshake growing into a trillion dollar network, so the security budget won't be that big, therefore I don't think it will be very energy intensive. Furthermore, if you calculate the economic impact of DNS hacks, the net impact of a decentralised PoW DNS implementation could even be positive.

Wrt to non-PoW system, so far governance for those chains looks closer to a federation (where a few agents control the majority of the network) than to a really decentralised network. In that sense, a proof-of-stake DNS network wouldn't be that different from the current implementation. If such network ever takes off, I wouldn't be surprised if major ISPs, Cloudfare, Google, and a few other players end up owning the majority of the tokens.

[1] Adam Back's 1997 Hashcash, designed to fight email spam and DDOS attacks was based on PoW.