These LLM tools appear to have an unprecedented amount of access to the file systems of their users.

Is this correct, and if so do we need to be concerned about user privacy and security?

We should be absolutely terrified about the amount of access these things have to users systems. Of course there is advice to use a sandbox but there are stupid people out there (I'm one of them) who disregard this advice because it's too cumbersome, so Claude is being run in yolo mode, on the same machine that has access access to bank accounts, insurance, password manager and crypto private keys.