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Nowadays, it's hard to get any where with out having some form of post-16 qualification, so yes, Harry is possibly setting himself up for a difficult time career-wise in the future. However, I think that he has other issues on his mind right now.
It seems like their post-Hogwarts education is largely apprenticeship-type training. I wonder if any of the muggle-born students have gone to a traditional university?

I believe there's a hero in all of us that keeps us honest, gives us strength, makes us noble, and finally allows us to die with pride, even though sometimes we have to be steady and give up the thing we want the most. Even our dreams. -- Aunt May, Spider-Man 2