I played "Magical Stone Excavation" and chose "Broww, huntsman of the dark world" as a discard from my hand. However, the draw effect from Broww did not trigger. The official text from MSE says "Discard 2 cards..." which means Broww should interact with it, since he has actually been discarded as opposed to just being sent to the graveyard.

Yes, it's a discard, but it's not a discard by a card effect. The discards for Magical Stone Excavation are a cost. Costs don't count for effects like Broww.