Hi,
This is a great question. And the first element is that I think you should (i) lower your expectation regarding staffing and (ii) think of the bigger picture.
Regarding staffing procedure. Staffing is quite complex. It's not simply staffing the next mission but actually arranging the pipeline so all the next X projects can be launched properly (with trying to optimize the situation for the next few months). So no, the staff cannot take the time to sit with every single consultant to get their need and put them on the right mission. More often that not, the partner will simply say I need X consultants for Y weeks and I would like this manager. The staff will then simply take whoever is free to fill the count and make the situation works with the other projects to launch.
Moreover, as a new recruit, you don't have the view to know what is good and what isn't. Even for people with years of MBB under their belt, it's not uncommon to think a project looks good on paper and turns out to be awful (because of the team, the clients, the duration of the mission, or key elements that are missing…). And the opposite is also not rare.
So what you should do:
Regarding staffing. You actually should connect with partners and managers. Learn what new projects will begin and if interested ask them to be part of it if possible. If you do a good work, they will gladly take you with them. Hell, you probably will not even have to ask. I personally think it's not a good idea for new recruit (because consulting advantage is the opportunity to discover new fields, new projects and new situations often, so narrowing your field of opportunities as soon as you join does seem contradictory) but to each their own.
Regarding working on your strengths and weaknesses. I think you are looking at the problem under the wrong angle. Most missions will have a lot of different components and with any mission you will have the opportunity to work on at least a few of your shortcomings. So in my opinion and from experience, the best way is actually to connect with the manager or the senior once you've received your staffing and ask for a quick discussion when he has the time. Use the opportunity to tell him the projects you've done (with who), the tasks you've took care of and the feedback you've received. Then mention your axes of progression and ask if it would be possible during the mission to be exposed to a stream that will allow you to work on that (or to discover a new aspect you've never touched so far).
Regarding staffing members, it sure is annoying to not get a response, but usually the best way is actually to discuss face to face with people. Discuss with them about the next missions and if you've already been staffed. Once you've receive your staffing always reply with a thank you mail for the new mission and so on. Even when you're staffed for the next few weeks, go ask what the missions for next week will be and stuff like that. People may call that networking or whatever but in my opinion, this is just being polite. If you only talk to people to ask for a favor of course, they will be less incline to help you.
(edited)