You will have to endure some degree of printer cartridge-type work in your early years, no matter what, especially since you're the new person. But if the role exposes you to the work you enjoy, put in the hours and you'll get noticed.
I remember my first years in networking - I never enjoyed climbing into ceilings, pulling cable etc. But I got it done, and worked hard in the areas I really enjoyed. Pretty soon an engineering role opened up; I was able to move into it quickly, as management had seen my potential.
You talk about "doing things that others have solved" - this is the "make vs buy" problem and, to a degree, it's fine to build things yourself. But you can probably appreciate that it isn't just about building the thing, it's about maintaining it. In a professional setting, you have to choose the optimal solution for the business.
On your personal time, build stuff! Whether that means assembling FOSS solutions or writing things from scratch, it's valuable experience.