Should I leave Corporate Strategy for Consulting?

Wanted to get your thoughts on this. I am post-MBA and have been working in Fortune 100 Corporate Strategy for 2 years now. I don't have traditional consulting experience, but have worked in client services before. I am wondering if I missed out by not obtaining the traditional consulting experience before coming into Corporate Strategy.

My biggest concern comes to developing the ability to manage projects on my own. In my current role, I report into a Director (the equivalent of an Engagement Manager/Project Leader at MBB). I am responsible for the analysis/modeling/slides and will push some of the work to an Analyst when he has capacity (he is a shared resource). I am currently struggling with learning to play the role of Director on projects and wondering if switching to a consulting firm for 2-3 years will get me over that hump. Sometimes I fear that I am just an overpaid/glorified Analyst and I am not sure if/when I will be able to make the leap to the Director level (Manager/PL at MBB) where I am able to manage projects. I feel like working in consulting will help for 2 main reasons: 1) high number of reps (projects). Although I am constantly busy in my current role, there is also a lot of BS work that needs to get done (ex. Board material etc) that won't really help me level up to the Director level and 2) the development/mentorship opportunities in consulting. Since everyone above you has pretty much gone through the same path before, I feel like it would be easier to develop in that environment

Does any of what I'm saying make sense? Long-term goal is to move into a role where I have P&L ownership so moving into consulting would take me a bit further away from that but I'm wondering if it would be worth it to do for 2-3 years. At a minimum, I feel like it would help from the confidence end.