What is the key difference between the Proof of Activity proposal and Proof of Stake? coblee (Litecoin creator) suggested a new Proof of Activity method, to complement Proof of Work, instead of using Proof of Stake (oh my!)

Can someone provide a tl;dr of what is the key problem in PoS that PoA is trying to solve, and how does it solve it?
From what I understand, Proof of Activity and Proof of Stake are methods of preventing a 51% attack.

Proof of Stake basically means that the people that hold the most coins sign the blocks that will be part of the block chain. This means that the people that have the most to lose from a 51% attack will do their best to prevent it and not risk losing their money. However, this also means that the rich users can control the network.

Proof of Activity from what I gather is a method of selecting a random peer from the entire network to sign a new block that would be temper-proof. This would mean that a 51% attack could not execute too well, as it couldn't predict who the signing peer be in the future.

The latter idea appears to be better for preventing attacks on the network, providing there are no logical problems in there. However, I have not studied either of the two methods in greater detail so I might be wrong.