There's a LOT to unpack here and I really don't think a Q&A can really do it justice! I genuinely think you should consider any of the following to help you sort this out: 1) Networking calls (you can be more honest here if you find them on LinkedIn for example), 2) A career coach
There's really a lot to both of your questions AND you've made a lot of points in your writing that I think need to be discussed!
That said, here is the long and short of it:
1. In what capacity can I work as a consultant?
In a large capacity. Ultimately in any desire you really choose. At BCG worked with an ex-flautist, an English major, a Dentist, and a professional violinist for goodness sake! Moreover, they were all better consultants than I ever was!
2. What kind of impact can I make?
Similar answer. If you have a dream and work hard/set your mind to it, you can do it at MBB.
3. The questions/answers you didn't ask
The day-to-day of consulting is extremely tough and often tedious. There is a lot of work spent on powerpoint, client management, oftentimes b.s. etc. That said, the big picture of it all is massive, and there is often huge impact.
You won't come in as an analyst but rather as a Consultant, or, possible, a Project Lead/Engagement Manager.
The higher your level the most impact you have, the more control you have over your direction, but also the more stress you have as well.
This is a lot to talk through...feel free to reach out if you'd like to talk through this in a career coaching session.