When I was in college, jumping through the hoops, they didn't believe that you needed much skill in the other instruments. We had to learn the basics of all of them, but not much more than that. By basics, I mean:
--How to put the instrument together
--How to form a proper Embouchure
--How to care for the instrument
--How to play on a 5th grade level.
I didn't agree with the last bit... We did however have to achieve a high level of ability on one instrument. Each semester we had to "jury" in front of the faculty. This is basically an audition into the next semester of private study. The final jury is a Senior Recital for the public. The school believes that by becoming virtuosic on one instrument, the skills can be transferred to other instruments for teaching purposes.
In my opinion, this is only partly correct. It is true that basic musicianship skills are the same across the spectrum, but the actual physical process of learning another instrument has its own unique challenges. For example, a Horn mouthpiece is very different from a trumpet mouthpiece and requires a good bit of practice to become proficient with.
My advice to you would be to learn as many instruments as possible and go as far as you can with each one. Don't settle for the basics. You're talented, see how far you can go with an instrument. I never settled for "Hot Cross Buns" on the Bassoon... I took a Rubank Method book and practiced as far as I could. I learned all of the scales on each instrument. I think it's made me much more well-rounded and prepared for teaching.
As for the jury process, I think Music Education majors should have to jury on a different instrument each semester at a high school level. I think it's important that they still study one instrument, because that is where the musicianship skills will be developed. Yea, it's more work, but it's worth it, if that's what you want to do.