I'll try to keep it short. When reading your summary it looks as if the root cause is a perceived lack of authenticity.
The biggest and most common mistake I see candidates making when trying to answer fit question is that they talk a lot of BS and I only hear bla bla bla as an interviewer.
One additional aspect might indeed be the pure "delivery" - i.e. how exaclty you communicate your answer (including voice level and intonation, body language, ...) which makes it not 100% authentic.
However, based on more than 10 years experience this is mostly only the symptom and not the underlying root cause. The real issue behind is that candidates are not able to draw a clear connection between them and the firm, and thus only talk about generic things that any candidate can say (like it's a great brand and intellectually challenging environment with top-notch C-level clients, bla bla bla bla).
The trick here is to establish a clear connection to yourself, how exactly you fit in into this picture (e.g. what's your proof to be able and enjoy working in such an intellectually challenging environment, etc.)?
Concerning cases - ABS! (always be structured). Even if the case is more conversational in tone, it's still a case interview in which you need to have a rigidly structured approach.
Hope this helps - if so, please be so kind to give it a thumbs-up with the green upvote button below!