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.
The faulty assumption here is that a lifting wing always produces induced drag. An infinite wing does produce lift, yet no induced drag. This is really a chicken and the egg question. The tip vortex, the downwash, and the induced drag are all part of one and the same phenomena - and of course a tip vortex cannot flow without spanwise flow, so throw that in there too. The downwash is simply the vortex flow considered along a vertical longitudinal plane. Yes, reducing the vortex is reducing induced drag.