How do I show genuine motivation for a consulting team in an interview?

It's hard to say exactly what you could have changed without hearing your answer. That said, I tend to find that a story/flow (i.e. a Resume Walkthrough but with a theme of "this all has led to me being perfect for consulting") tends to work best.

If you listed a memorized list of 3 reasons why consulting, this can often come off as bland and disingenuous.

Honestly, the best way to improve is to get feedback from someone! Give your "why firm/consulting" answer to a family member, friend, or coach, to figure out how to exactly have that "umph" to your pitch!

For me personally, I talked about the things I loved + learned as an IT Consultant (i.e. problem solving, teamwork, leadership, client management, diving into many industries etc.) AND the 1 thing that was missing that drove me to wanting strategy consulting (i.e. wanting to have impact/control at a higher level, in regards to figuring out if an IT system was even needed, how the organization needed to adapt/ajdjust, strategic/cultural decisions/changes, etc. etc.)

You need a similar story :)