Ultimately, you will just have to do what the interviewer tells you to. However, I think your instincts are right!

With anything numerical, there are typically 3 stages:

1. Verifying objective: When you are presented with a bunch of data, numbers, own estimates, etc. you will have to decide what to do with it. This is more challenging in candidate led cases, where you're “in charge". In that case, consider overall case objectives, your framework (don't just start random calculations), and what numbers are available. Then verbalize what it is you would like to calculate and why. Not sure if you did that but you should (unless this part is explicit in the case e.g. client wants you to calculate a payback period for investment x).
2. Verifying approach: Once your interviewer is on board with the objective, you need to figure out how to get there. That's basically your equation, which ideally includes variables that you have data for. Verify with the interviewer by verbalizing top down. If he wants you to just get on with it, that's what you'll do. If he pushes back, keep composed and explore what's wrong (that's why you verify in the first place instead of going down the wrong route).
3. Execute: This is where you can go silent once you politely ask for some time. I never expect candidates to talk me through the actual number crunching because it's not effective. Maybe communicate some partial results to break the silence from time to time. Other than that, the objective is to be quick and correct, not chatty.

Hope this helps figuring out you own strategy! Best of luck!