Does it make sense to design a mutable blockchain? I work in a large consulting firm which is marketing a mutable blockchain solution to its clients. But I personally fail to grasp any advantage of using a blockchain design if you are discarding immutability. Am I missing something here?
I agree more context is needed to provide a more directly meaningful answer. 

But in general: a blockchain allows users to interact without trusting a central authority, instead they verify the system’s state on their own. As a user, I can verify that I own some amount of bitcoin by running a node, and seeing the transactions that paid me as confirmed in the blockchain record. I know that the network’s rules do not allow those coins to be moved unless a transaction is signed by the valid keys (which I control). If the system allows mutability, then how confident can I be that the transaction which paid me will not be changed in the future? With an immutable blockchain, the energy spent on POW mining helps guarantee this, but if mutability is allowed... ??? How can I be sure? 

If some form of mutability can be added while preserving this factor of trustlessness, then maybe a ‘mutable blockchain’ is reasonable. But I am yet to see such a proposal. If the user must trust a central authority to not change the system, then using blockchain architecture is superfluous. 

In reality, I suspect in most cases a ‘mutable blockchain’ would be an inefficient way to implement a database, albeit one that can be dressed up in marketing buzzwords and sold to customers more easily due to the ‘blockchain hype’. 


  But I personally fail to grasp any advantage of using a blockchain design if you are discarding immutability. Am I missing something here?


Advantage: customers love blockchain hype, so they’ll be more willing to buy what you’re selling. 

I’m sorry to say, but I think your intuition is correct. Frankly, such projects seem either misguided, or scammy to me. 

EDIT: perhaps worth mentioning as well: allowing a central authority to alter the blockchain record creates a monstrous security risk. No matter how well intentioned or benevolent the authority is, an attacker targeting their system of control represents a huge existential risk to the network. There is no such thing as a secure backdoor that only ‘the good guys’ can access.