Sure, but by the same reasoning, you don't need to change your heading by more than a fraction of a degree before the interceptor has to massively change it's i

Sure, but by the same reasoning, you don't need to change your heading by more than a fraction of a degree before the interceptor has to massively change it's intercept trajectory. A scramjet powered missile randomly making slight adjustments around the desired trajectory would make it extremely difficult to intercept.