So, mainly what you want to ask, is: is it possible for one be competent in both design and coding. That is, produce good results.
Well, it is. There are people that are good at both, and have launched products with success.
I said good, not excellent, and here's why:
Nobody cares about actual "excellent" in a startup.
What you should care about is "good enough". Excellent is the enemy of shipping stuff. If Facebook tried for excellent, it wouldn't be built with messy PHP code and butt-ugly graphics when it was released. If Linus wanted to release something excellent, he wouldn't release a half-baked implementation with tons of problems, drivers missing etc in 1992. You get the point.
I also find advice like "You need to focus on ONE of them", "excellent" etc, non pragmatic. It's pop advice acquired by reading too many bullshit business self-help books. The reality is, especially in a startup, people have to wear many hats at one.