Distributing it gives freedom of choice. No need to nominate a blessed CA that everyone has to use.
It's fine for anyone to mine cheesecoin. Probably better for this specific trade group in fact, if nonmembers decide to use this same blockchain for their cheese as well! Customers can still check if they are buying DOP specifically vs. 150mi-local worker coop biodynamic or whatever. The more widely used this system is, the more wholesalers, merchants, and consumers will learn of it and use it, and the harder it will be for fraudsters to pass off their goods as the Genuine Article.
Who's going to run the CA? Who's going to pay for it? Can you get all the customers to add the CA's public key to their trusted list? Will all the producers and merchants agree as well? The producers are the easy part, they are the ones coming up with this plan. Everyone else in the cheese economy: that could be harder. It's not just verifying the "this cheese was made by X" claim, also at every step of distribution you have to update the log "this cheese was sold to Y" and get that signed too, so all the distributors and merchants need to get certificates and learn how to apply the secret keys to cheeselogs. Using a blockchain also requires some crypto-effort, but it requires participants to agree on fewer things, gives the participants more freedom of choice. I could imagine someone proposing the centralized version you suggest, but apparently this particular group think it is better to do it in decentralized fashion and I can definitely see some advantages to that.
