Does pistachio ice cream need pistachio nut chunks in it?

I was at first a bit uneasy about your question, seeing the word "need" as potentially subjective. Then I realized that the core of your question seems to be:

Would the cream part of the ice cream (what I really like) have such a different taste if I made pistachio ice cream while skipping the "pour in pistachio nuts" step

No, there will be no different taste. The flavor doesn't come from the chunks.

To give you some background: Ice cream is made in following steps.

• Mixing a liquid base. It is what makes the "bulk" of the ice cream. Each parlor tends to have its own recipe for the "base", which usually gets made in large amounts and refrigerated.
• Flavoring and coloring the ice cream. Depending on the flavor and the parlor's recipe, this can be natural or artificial flavoring, or the name-giving ingredient in a pureed form, or a combination. For pistachio ice cream, if you want the real deal, it is pistachios ground to a paste, analogous to peanut butter. Specifically for pistachios, it is economically unfeasible for a parlor to sell ice cream made with nothing but real ground pistachios.
• Churning the ice cream. The flavored base gets frozen under constant agitation, to achieve the desired texture.
• Layering the ice cream. It gets scooped out of the churning machine and placed in a container. At this stage, manufacturers can add all sorts of mix-ins, either solid ones (like your pistachio chunks) or as a syrup. This is only a kind of embellishment, the ice cream itself doesn't change after churning.

As you see, the pistachio ice cream is ready, flavor and all, when it comes out of the ice cream making machine. The chunks are additional. Thus, by your definition for "need", the answer is clear. Pistachio ice cream doesn't "need" pistachio pieces in it.

Short answer: no, it doesn't.

I'm actually a bit surprised by the other two answers here about (pieces of) nuts being a common ingredient. Where I'm from, pistachio ice cream usually does not contain pieces of nuts, regardless of whether the flavoring is artificial or natural. (fun fact: pistachio ice cream is usually a good indicator whether an ice cream shop is any good, because you can tell by the color if the flavoring is artificial or natural)

I think adding things like nuts, chocolate, cookies, candy, etc. does two things. It adds a different texture to the ice cream, so you get a crunch or a chew or both, in addition to the smooth texture of the ice cream itself. And usually, it adds flavor when you bite into one of the pieces of chocolate or whatever you've added. Some things like chocolate add a lot of flavor when you bite into them, whereas I think things like walnuts are more about the texture. Pistachios do tend to have quite a bit of flavor in them, so yes, I think some of the flavor would be lost if you didn't put them in - and of course, the whole textural element as well. But hey, if you're making the ice cream, you can make it any way you want. ;-)

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.

Side Notes:

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.

It wouldn't be pistachio ice cream if it didn't have any pistachio in it, even it is just for show as most of the flavor is probably artificial.

For example, the Coaticook pistachio ice cream (local ice cream in Québec) is made from natural and artificial flavors and also contains some pistachio pieces.

Most of the recipes I've googled for use almond flavor as the main flavor in the ice cream.

I assume most commercial ice cream follow the same kind of recipes.

I also assume smaller artisan ice cream makers will do things differently.

Anyway, do as you wish, you are the Joe Di Maggio of your pistachio ice cream (to quote Chef John).