It is trivial to create such a blockchain and you won't need a centralized website to see the messages, the full node can do this very easily by parsing the transactions and showing the message inside them to the user. Even a light/SPV protocol could be introduced for those who don't want to run a full node.

However, the problem is that you won't be able to keep the blockchain alive in secure and decentralized way. Simply because it lacks incentive.
For the ledger to be distributed and decentralized, people have to run full nodes which means they have to download and verify and store  the entire blockchain which is essentially lots of messages they don't want. If you add the "token" on top of it to create an incentive then the project stops being a message platform and becomes a payment platform where you can't post a message for free like you do on centralized alternatives like Twitter. It also adds the problem of "price", if it dumps nobody would mine that chain and the PoW algorithm becomes vulnerable to 51% attacks.