If "Draw three cards" and "Discard three cards" are listed as a single ability, no one has priority between the two actions, and therefore instants and activated abilities can't be played.

Here's a few examples where your opponent's argument doesn't work:

[[Burning Inquiry]][[Frantic Search]][[Rhystic Scrying]] (assuming someone payed 2 against scrying)

And supposing that you played a blue-black creature, and someone redirected the discard ability to you, the counter example is:

[[Dire Undercurrents]]

Again, the only difference is that Undercurrents has two distinct abilities. Since instants and sorceries without replicate or copies put only one ability on the stack, your opponent's argument isn't valid.