First, I suggest dropping the "buckets" approach. That is boiling the ocean.
Second, you have to approach problems not from generic POV, but try to solve the specific problem at hand. You are asking about a framework for a problem as generic as "PE wants to acquire a company". To solve a problem... you need to know what the problem is. And to know that, one needs to know 1) Context; 2) Objective / Specific Question requiring a recommendation; 3) Success Metrics
Third, you have to consider what would be a real life interaction between a client and a consulting firm. In this case, if a PE approaches you and asks "should I buy this company", would you reply "well, first I need to understand whether you actually have financial capabilities or the financial expertise to do it!". I believe the conversation would end immediately and they would look for another advisor.
Moreover, we're discussing strategy here - and the question of "capabilities" very often is misunderstood as a matter of operational capabilities - which goes into the realm of implementation. To make a parallelism... it's like asking if one should practice for a marathon and someone replying that the may not have some tennis shoes. They may have them or not, but it's completely irrelevant for the question at hand. First you decide whether you should run the marathon. Then, if the answer is yes, you discuss the execution, i.e., you go and buy a pair of shoes. Whether you have them or not is not driving the answer.
Same applies here. First you decide on whether it is a good idea. Then you discuss how to executive and overcome any existing limitations.
Exception being... if they upfront mention a limitation they have. E.g. "we don't want to buy companies above a $500M valuation." Ok, then you test that. Not in a "bucket", but as a specific analysis / driver!
Of course, you may be thinking instead of having expertise in order to find synergies and quantify them. Ok, but in that case, what you have is not a "capabilities" bucket. You have a specific assessment to quantify synergies (do you see know why "generic buckets" don't work? They don't really show how you solve a problem, they do not provide clarity to the interviewer, and they just suggest you will boil the ocean and explore a ton of topics before getting into an answer.)