I think the point of the explanation was to provide some intuition as to where the Exp[- 2 pi i k n / N] in the formula comes from. If it's easier to think of that as a sinusoid then your way of thinking about it is better. If it's easier for you to think of that as spinning your signal around like a phasor then the OP's explanation is useful.

I think that rather than explaining what a Fourier transform does the linked page attempts to explain how it works.

I agree with you. I know the Fourier transfom well from a discrete signals class but that explanation didn't make any sense to me. What is the motivation to spin the signal at 3k?

Because that's the frequency component you want to measure. Repeat at all the other frequencies you're interested in to get the whole spectrum.