While I agree with your sentiment, studying abstract algebra at the undergraduate level we never got more than a broad sketch of a proof of the general insoluability of the quintic. It more seemed an opportunity for the professor to tell the life story of Galois -- mathematical genius and political firebrand killed in a duel over a woman at age 20.

That being said, it is difficult to find a motivating example for much of elementary abstract algebra -- which is doubly hard, because for many students abstract algebra is a new level of abstraction, math without numbers and it takes something to make that leap.