Good call on the trailing vortices... vortex shedding is indeed a different phenomena. Trailing vortices are what I meant, just not what I wrote. (Oh surrrrrre, he says. Sure they were!)
Spanwise flow is simply not required for induced drag, although it is almost certain to occur for a lifting finite wing. To prove this to yourself, just ask whether a wing with no spanwise flow could generate lift, and consequently generate induced drag. The answer is yes.
We gonna have to agree to disagree! LOL
Don't forget that spanwise flow is not limited to being
on the wing planform. Most of it can be aft of the wing. Case in point... engineered tips (that is, engineered to control spanwise flow) don't make too big of a deal on a supersonic wing -> e.g. just about any supersonic modern jet fighter. This is because there is little to no induced drag in supersonic flight (at least not this type of induced drag, supersonic flight also has what is called "wave drag due to lift", but it's not the same thing).
When low speeds are important (and induced drag is at it's highest) this spanwise flow becomes important... e.g. the drooped tips of the A-10. Or Hoerner tips on some GA aircraft. It's really just a judgement call on the part of the designer... you'd have to do some tests to prove that an engineered tip is helpful for a particular configuration, depending on the aircraft's purpose/mission.