>The orthogonality is essentially follows from (1) integer frequency complex sinusoids have an average value of zero over [0,2π], and (2) if you multiply two distinct integer frequency complex sinusoids, you get another integer frequency complex sinusoid. I'm not sure that this is any more intuitive.

i think these kinds of explanations are hilariously pointless. and i don't mean to disparage because you're just trying to answer op's question but all you've done is restated the proof in english - i.e. of course it follows from that because what you've just said is the inner product of basis functions is 0. well yes of course that's definition of orthogonal.