Probably why I am not a game dev but when the author mentioned and discarded the iterative solution in favor of the closed form solution. I immediately thought how cool it would would be to have some sort of deep involved simulation that would in the end create the same effect.

One reason a missile might spiral like that is in order to have a simplified seeker head and control assembly, you only seed to seek and control on one axis and you spin the whole missile to scan over two dimensions. the missile will always overshoot a little but correct once it rotates around the other way. while this makes for a far simpler seeker and control setup it wastes a lot of energy so you need a bigger engine for the equivalent range if you had a two dimensional seeker/control missile.

Another reason is to make the anti-missile solution much harder. It still wastes a lot of energy.