Long story short. Don't put all your eggs into one basket.
... Whoa. Well, that makes sense. Like Kurisu said as well, it seems like they look for personality types, even if those end up being hit-or-miss in the end. I don't necessarily plan to make this huge all-star resume and then expect that to coast me through everything, but I still do think that it would help me a lot to just get some teaching experience in general. I'm relatively shy in person, so it'd be good for me to know how it feels to stand in front of a class and actually do something.
But I will try to prepare from here on out without focusing too hard on the technical details when it comes to teaching and all. I'd like the experience prior to applying, but I won't make it the main focus, I think. Thank you for your insight and advice, I really, really appreciate it!!