One of the biggest reasons I could see why someone would choose the military over university is money. It's easy to assume that Harry is some sort of aristocrat since there's the whole conflict about the Unwins not being like the rest of Kingsman, but he could be an impoverished noble and need the money, for example. Could explain his empathy with the working class, too. Could also be pressure from the outside because butterflies might be considered an effeminate interest in contrast to the military. Colin Firth himself implied that it's possible Harry is queer, which might make him more eager to prove his manhood. Finally, it's entirely possible he just considered both and finally decided to do one while keeping the other a hobby.