Does pistachio ice cream need pistachio [chunks] in it?
No! You can absolutely follow a pistachio ice cream's base/custard recipe while skipping any final "pour in pistachio nuts" inclusion steps and still get the flavor you're looking for in the cream.
The (common to North America) example of B&J's "Pistachio Pistachio" flavor is a combination of a pistachio ice cream base with the addition of pistachio nut chunks as an inclusion. In pistachio-on-pistachio combos like that, the inclusion is there primarily to add a textural element whereas most of the flavor comes from the creamy base.
In typical recipes, most of that base flavor will come very early in the custard making steps from the addition of finely-chopped/ground pistachio nuts (or an extract, artificial flavoring, etc.) to the ice cream base. This part is absolutely necessary for that pistachio flavor to get into the cream of your ice cream.
The second pistachio addition of larger nuts/chunks, usually done just-before or just-after running the custard base through the ice-cream-maker (depending on device recommendations), is an optional step for the textural inclusions. Skipping this step should only really affect the mouthfeel of your finished product, rather than having any effects on the flavor of the cream itself.
As a fellow pistachio-cream lover, good luck! I do still recommend trying (room temperature) nuts/chunks as a topping, though, because in that preparation the texture is more light/crunchy/crumbly rather than the hard/waxy/chewey texture that frozen pistachio chunks tend to give. (Pecan-chunks-topping-pistachio-cream being my personal favorite thus far.)
Also, if you have an Asian grocery store anywhere nearby, consider trying some Matcha/Green-Tea ice cream. It's definitely not the same thing as pistachio, but the "muted vegetal alongside vanilla" vibe is pretty similar.