Why are even number of point sources considered to explain Single Slit Diffraction Patterns?

The problem is, taking an even or odd number of point sources is an approximation and therefore seems to lead to discrepancies in any case; as the guy says in the video, one should take into consideration an infinite number of point sources, but drawing them would be too long so he chooses to take eight.

The reasoning comes from Huygens' principle (what is it?): so to be completely correct, one should make all the calculations with an integral (thus including every one of the infinite, infinitesimally small point sources) and would indeed arrive to the same result for the interference pattern, obtained with rigorous means.

So to answer your question, the need to have an even number of point sources is a consequence of the approximate nature of the reasoning used in the video, and is not an intrinsic inconsistency of the theory (when you have an infinite number of point sources, it doesn't even make sense to wonder if they are even or odd).

To really discribe single slit diffraction, we must assume, that every point of the slit acts like a point source. So really there are not 8, 9, or 10, but uncountably many of them, and the notion of odd or even does not make sense. We must add (integrate) contributions from all this points, and baceuse there are infinitely many of them, each contribution is infinitesimaly small. So choosing finite number of points is only an aproximation. Nevertheless if you would choose an odd number of points, than there would still be some places on a screen, where the intensity is equal to zero. This would be the places, where all the contributions, from let us say 11 sources add up to zero. In this case, there would not be this pairwise destructive interference you mentioned. But if you have three sine waves, 120 deg out of phase each, they will add up to zero.