Of course not. Even setting aside the question of how primitive the cultures in question are, there's really no such thing as a primitive language.* Every language expresses precisely what its speakers need to express. A language spoken by a nomadic tribe of herders and hunters might not have technical words for scientific or sociological concepts, but it'll have vocabulary that enables people to talk about a whole range of concepts, emotions and relationships that are important to them — many which can't be clearly expressed in English or other majority western languages.
The "heap many moons" style of speech is a very simple example of pidgin, a type of language that arises when people from different cultures need to communicate on a basic level. It's usually for trade, but pidgins can be heard whenever Britons or Americans abroad are trying to make themselves understood to "foreigners who have the nerve not to speak English".
Where the contact is regular and long term, a pidgin can develop regular rules and vocabulary, and eventually, given the right circumstances, children might start growing up speaking nothing else. At that point, it goes through a metamorphosis into a creole, a language that's flexible and rich enough for its speakers to say whatever they need to. There are creolised languages from the Caribbean to the Pacific, and many are elegant and expressive.
All languages intended as the primary means of expression for a people are tailored exactly to what that people needs. Whether or not the Inuit really have fifty words for snow,** they can certainly talk about snow in a lot more detail than a people whose language has evolved on the equator.
What a language does or doesn't have inevitably reflects what matters to the society. Many of the Australian Aboriginal languages didn't have counting systems at the time Europeans first arrived. This wasn't stupidity — if you rarely see more than a handful of any given object, including people, why would you need to count? As soon as the concept was introduced to them, it took a remarkably short time for this gap to be filled.
On the other hand, many of them have degrees of sophistication in their grammar that European languages can't match. English, for instance, has one way of expressing the first person plural pronoun — we (with the variants us, our and ours). Some languages, though, (including Old English) distinguish between whether you're saying I and you or I and they.
This might seem strange to those of us whose languages have done without it, but it's actually a distinction between two very different concepts. If you tell someone "We're meeting at eight o'clock," you might mean "We're meeting — you can make it, right?" or "Us lot — we're meeting up. Just saying." In English, we have to rely on tone and context to make it clear which we're saying, but it can be a useful distinction to make.
On the other hand, in many of those Aboriginal languages that didn't have counting systems there might be up to a dozen different ways of saying we, depending on exactly who the other person is, whether they're related to the speaker, whether or not they have the same Dreaming. In these societies, it's vitally important to clarify these issues, and the languages have developed incredibly complex grammar to accommodate that need.
A similar, though simpler, concept that occurs in many European languages is the distinction between the familiar and formal versions of the second person pronoun. This will be known to anyone who's learnt French, German or Spanish. In French, for instance, you'd address a close friend or family member as tu and a more casual acquaintance or stranger as vous, while German has equivalents for both the singular and plural forms.
English used to make this distinction, too, using thou and you, but thou has died out, except in a few dialects. There are various theories for why this should have happened, but the effect (if not the reason) is that English-speakers don't have any of the complex social niceties needed to use this particular grammatical form.
The history of language is littered with abandoned grammar that once expressed vital concepts. Early forms of the Indo-European language family, which includes almost all European languages,*** had not only a singular and plural, but also a dual number. This may originally have been used to express any two things, though by the time it reached classical Greek it was only used for specific pairs: the eyes, the ears, egg and bacon, Simon and Garfunkel and so on.
On the other hand, the abandonment of the dual may reflect a fundamental change in how we view the relationships between things. For us, there's an obvious difference between one and all other numbers, and that forms an essential part of the patterning of our minds. There's a mathematical justification, of course, since one really does behave in ways that are different from every other number. On the other hand, two is also a unique number, the only even prime, and viewing doubleness as a thing apart in the same way as singleness may have been integral to how those societies saw the world. Perhaps it explains why triple deities are so common, if three was the first plural number.
What a language does or doesn't include can have an enormous effect on a society. To return to classical Greek, there was a simple but far-reaching linguistic habit among the Greeks. The language had two little words (men and de) that could each be slipped in as the second word of a clause to set it up in opposition to another clause or sentence. You could roughly translate men as "on the one hand" and de as "on the other hand", but such little words could be used without the clunkiness of the English phrases.
It's unlikely to be a coincidence that the language which adopted this structure was spoken by a people who essentially introduced philosophy and logic to the west.**** Whether the structure nurtured a logical frame of mind or the impulse for logic created the structure (or a bit of both), a naturally dialectical language was perfectly adapted for Socrates, Plato and the rest to debate philosophy.
From subtle relationships to logic to high-tech (or even talking about snow), all languages are rich and expressive in the concepts their speakers care about — and, as long as they're human, that will certainly include a full suite of emotions and imagination.
If the hero(ine) in your story meets a primitive tribe, by all means show communication difficulties between them, but don't make the mistake of believing that really is how they speak or think, any more than your hero does. They're probably too busy gossiping about the stranger who doesn't know the first thing about their way of life to bother with all those heap many moons.
** I'm fairly sure I've come across a debunking of that, but I'm not certain.
*** Except for Finnish, Estonian, Lappish, Hungarian, Turkish, Maltese, Basque and some minority languages in Russia.
**** It's often said that the Greeks "invented" philosophy. Of course, the Chinese and Indians also "invented" it, and no doubt other cultures did too, but however dubious that claim might be, the Greeks certainly originated the western tradition of philosophy.