Ok so here is my 2 cents
If you are not going to deal well with an injury or blood/guts, being a stable hand/trainer/prof rider is not a good ave for you(in my opinion). The reason i say this is because any of those jobs you are expected to be able to take charge and act quick in a situation where a horse is hurt, cut themselves, broken a bone, crashed through a fence, got into it with another animal - you get the point. I am not saying that you couldn't do any of those things BUT if you don't think that you can have a quick reaction and ignore your "squimish-ness" then it would probably be best not to venture there. Clients are going to expect that you are going to act fast and be effective if something happens to their horse(s) when they are not around- it comes with the territory
Working for an organization sounds like a great idea! If this is something that you want to pursue, i would suggest thinking what kind of a job you want to do with them- ie Marketing (planning events, fund raising, advertisement etc), Finances, Business Development .... This way you can focus your skills that you are gaining in college in the right direction. If you aren't sure what job you want to do, go talk to some people at a few organizations and find out what type of jobs they have there. I feel that if you focus your studies on skills that you can use in that job- you will have a leg up ahead of other candidates b/c you have focused on skills that will better the organization or company. Then you can still work at a local barn as a working student or volunteer at the organization so that you have another foot in the door when you graduate.
As for the nutrition- I know that MANY schools have a general nutrition degree for humans, i am sure that there is something that you could find for horses- but again remember that you might see some horses that will trigger that "squimish" button of yours. I say this b/c most nutritionists will probably work in co-hurts with a vet. (I would think but am not 100% sure) .. if you wanted to work with a feed company on their nutrition then maybe go talk to someone at the feed company and ask them what they are looking for in an employee doing the job you want.
Saddle fitter- this just sounds so cool to me but I have no advice on it
Anywho i am a firm believer that you can get a "practical" college degree and use it in the horse world. Just have to talk to some of the people where you want to go and see what they say. Oh and intern while you are in school (at a barn or organization or feed place)- you will learn the tricks of the trade!
Best of luck!
P.S. I am 26, have my degree in Business Management and once I pay off my student loans I plan to own a barn. Right now I work as a project manager for a company, still train with a trainer and teach lessons on the weekends- I am gaining all the experience I can to better prepare myself for owning my own place. So you may not get the perfect job right out of college but you have a lifetime ahead of you to plan it out!