The NYT apparently thinks that "curry" is a spice.
While curry itself is a sauce, I thought curry powder is considered a spice mix. "Cooks who can usually name every spice in a restaurant dish can’t smell curry or garlic" would presumably be referring to the spices that make up curry (as a spice mix), and not necessarily calling it a specific or singular spice.
Not really. Curry leaf isn't an essential (or even particularly common) ingredient in curry powder, or in curries. I think it's unlikely that the article was intending to refer to the odour of curry leaves specifically.