If I understand it correctly, you play by putting your moves on the blockchain here.

That’s the play by Twitch Chat concept ported to a blockchain?

What advantage does this have besides the moves being recorded on the blockchain? It’s more expensive and it’s slower (Cardano has 20 seconds block time IIRC).

Nothing is running ON the blockchain. Cardano doesn’t have smart contracts yet. You could exchange the blockchain part with any component that allows people to comment. For example Twitter.