I feel your pain... I think you'll find this feeling crop up a few times more in your journey as a designer if your experience is anything like mine. The only way through it is to stick with it, I often think design is more about the way you think and the way you approach challenges, if you are naturally falling in to this role in situations then I'm guessing that's due to how you think and approach a problem or brief?
All the other skills follow that intitial thinking, and utilise it throughout the process, in my opinion these skills are easier to learn than to change the way you think, so practice, then practice some more until all your other skills are where you want them to be, I know it's obvious, but it's true! (I'm pretty sure you'll continue to work on them through the rest of your career?!). So keep your head up and knuckle down (is that an oxymoron?!).
As for the connections, if you don't have them, then why not make them? And work on the things you want to share with them, they won't give people jobs just because they know them if they aren't any good. Design will always be competitive, I'd imagine there's one or two areas or points in their career that most designers have struggled, but I think it's worth the struggle. I started University without any background in art (not even at secondary / high school), and it felt like everybody else had even done an additional foundation year between school and university that I didn't do. I didn't think would be as big of an issue as it turned out to be in my first year... but I practiced, and I nagged uni friends to spend time with me and teach me what they already knew, I lurked on core77, and occasionally posted when I had the guts. It was a real confidence thing for me, but I think all the hard work paid off, I've reached a place where I use sketches to convey ideas more than I do words! I'm still learning, and still posting on here, but where before it was a struggle, it's now something I love doing!
If you love it, keep at it and it will all pay off.