What does having mutable state have to do with being decentralized?

Mutable history is the problem. What's written to the blockchain is supposed to be immutable forever and independently verifiable by everyone. If an entity can rewrite history (eg drop or revert transactions), it becomes a problem. This is what happened after the DAO hack and it resulted in the Ethereum classic fork.

Well, sort-of?

The chain still includes the transaction history of what happened with the DAO, but the hard fork added a special case unique state change which undoes most of the effects of the transactions that happened with the DAO.

So, the transactions still are recorded as having happened, but the state was edited?

Not that that necessarily is a particularly important distinction, but I think it is one that can reasonably be made.

The decentralized element should be more than just a software concern. Regardless of decentralized / mutable state if the control of the state of the chain (forks etc) is a centralized concern by a small group then we're just admitting the conceptual model is bullshit and this is literally just a very mediocre technical implementation of stuff we already have.

You can't just throw out or dismiss the historical failures of the model because it doesn't fit the ongoing narrative. The DAO was a glimpse at how easy the theory is corrupted, whether it started for the right reasons or not.