This happens sometimes when the interviewer has doubts about the candidate going in right path, this sometimes has something to do with the candidate behavior and sometimes don't (as it is very early in the case) - the interviewer may have more of a impatient/control profile.
Some Insider Insights here: The interviewer has a limited amount of time to evaluate you as a candidate in multiple dimensions, letting you run 2 mins by yourself and having the risk you might come up with a complete wrong framework is real (e.g. forcing a framework, understanding wrong the objectives of the client in the case, etc)
So what can you do about it ?
1. Always ask clarifying questions, focused on the context of the client and most importantly on the objectives (e.g. what problem are we trying to solve, what is the time horizon, how does success look like (KPI's to improve)) - this is an easy one and seems you've already done it
→ This shows proactivity and sends a clear message that you are on top of things (you are eager to solve the problem for your client in the most effective and efficient way)
2. Give an overview of first layer of the framework to your interviewer before taking the 2 minutes (e.g. in a public education case, looking for which factors you would analyze to improve it in X country … you can say “before taking a couple minutes to structure my thoughts, let me share with you the overall approach i'm thinking of, ok ? … i will analyze this main categories: educators (teachers), learners (students) & supporters (family), schools, government and methodology) … sounds good ? … great, give me a couple minutes to deep dive into those and comeback to you”
→ This shows confidence, great communication skills and empathy with the interviewer - including them on your plan before investing time on it (this is relevant in the day to day of a consultant, with your manager, with the partner, with the client)
* If the interviewer says it doesn't want this preliminary first layer, thats ok, no problem, you still showed proactivity and desire to communicate/align on work !
Of course to execute on (2), giving the first layer of your framework immediately it takes practice, first prioritize quality in your frameworks and then speed.