Actually that's how LastPass works (they move around an AES-256 encrypted database, and decrypt it on the client/browser).
The problem LastPass has, is that they re-use the same master password for two distinct things:
- Authenticating to login to your account.
- Encrypt your password database.
So in situations like this the loss of the authentication hash is relevant. I'd prefer to have a different password for the account than the database, but they don't offer that as far as I know.
In general I am broadly happy with LastPass's security. But it could be a little better for power users.