Am I just not cut out for recruitment?

If you got positive feedback from consultants and partners, then it sounds like you have your answer.

On one hand, I find myself enjoying the work done within Consulting.

Good, I still enjoy the job too. If you can do something you're interested in, you're in great shape. 

Yet, I find myself fixating a lot of on the doubts and obstacles. For one, I have poor self-esteem (it's something I'm actively working on),

Everyone thinks this. It's called "imposter syndrome".

I will never be 'smart' enough for Consulting.

Thankfully you don't necessarily have to be that smart for consulting or business in general. Work ethic, focus, communication, commitment to learning and improvement etc. goes further than raw intelligence. 

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In general though, I'd suggest you cut this line of thinking off hard. It seems to me like you're here for affirmation, and that's not particularly healthy. We can't judge your work, or judge how good a fit you are for the job, or judge how far you'll go - and you shouldn't limit your career and your confidence and your aspirations to whether you're "good enough" for an MBB consulting job much less whether some people on an internet forum think you are. Not to say you should just assume you'll only ever succeed. Be realistic about your options and know what alternatives you'll pursue if things don't work out. 

A couple of suggestions:

• Ask yourself what you're going to do with an answer to your questions here. Are you going to just quit trying if someone on the internet tells you you're not good enough? Maybe drop out of recruitment? Are you going to suddenly snap to a realization that you're good enough and become infinitely confident if someone here tells you that you are? The answer is presumably neither. So pursue what you want to do, and if you think you're falling short, figure out why and work to improve. If you do fail to get the job, then try for the next tier, or something adjacent. But don't self-limit simply because you're worried you aren't good enough - that'll just be a self-fulfilling prophecy.
• Get a legitimate personal opinion. You worked with people from MBB, you said they gave you positive feedback. Reach out, find some time for a chat or to grab a coffee, ask them their thoughts on your work, your potential, your weaknesses, your opportunities to improve and their advice on how to do so etc. You'll get nowhere spinning through hypotheticals in your head. Get concrete input from people you respect and who know you.

I also find myself being very drained from all the various activities I'm involved in to try and get my application to be as appealing as possible. Trying to stay on top of classes, competitions, clubs, networking, a social life etc is a lot. 

Sounds like you need to drop some things and prioritize what's important to you. Won't get any value out of all of this if you burn yourself out. "More" is also rarely simply "better".