The question is whether or not cultures that had no writing systems were limited to the same level of intelligence and/or development that other primates exhibit.
Given that with exception of an interesting knot-encoded (Quipu) system for some period of the Inca empire, the entire human population of the Americas (at least 15-20% of the total human population) fits this description, the answer seems to be "no". These cultures built huge cities (among the largest in the world at the time), used sophisticated irrigation schemes, ceramics technology and more.
Just to be pedantic, it is unclear if the Aztecs had "full writing". The notation system in the codices is probably more like a kind of mnemonic and accounting system along with a calendar. They probably couldn't write arbitrary sentences. But so little material survives we don't actually know!
The Maya though absolutely had "full writing" at least during the classical phase around 200 - 800 AD, and their writing probably derived from the older Olmec system. The Maya system worked much like classical Japanese or ancient Egyptian, and uses a mix of ideographs and purely phonetic symbols. They could, and did, write anything they could say. There are historical accounts of battles and names and etc. given in monumental inscriptions. And it is almost certain during the classical era that the Maya had full literature, with stories, mythology, histories, anthologies, religious scripture, etc. There were probably still priests who could read the Maya system when the Spanish arrived - one Jesuit recorded notes that indicate some of the phonetic values of some of the symbols correctly! - but they were mostly killed and most of the codices burned by the conquistadors.
It was not deciphered again until the 1970s and it's sadly still not widely appreciated that Mesoamerica had literate civilizations.
Fun fact: The Maya were such good astronomers and record-keepers that we have exact date alignment, which we have neither for Ancient Egypt or for Mesopotamia. For example any dates in Assyrian records are subject to a ~40 year window of uncertainty for exactly when they occurred. There is no such ambiguity for the Maya: Pacal the Great almost certainly died on August 29, 683.
Yes, for sure humans are fundamentally more intelligent than other primates, but still it does seem that writing and written communication is a big capability unlock.
It's impressive, and perhaps a bit surprising, what a culture like the Incas were able to achieve using only (we assume) oral tradition and passing down of skills, but there must be a limit to the complexity of what can be passed down this way - more along the lines of the skillset of an individual man being passed to an apprentice, but presumably also "managers/planners" passing down skills as well as tradesmen.
Given that with exception of an interesting knot-encoded (Quipu) system for some period of the Inca empire, the entire human population of the Americas (at least 15-20% of the total human population) fits this description, the answer seems to be "no". These cultures built huge cities (among the largest in the world at the time), used sophisticated irrigation schemes, ceramics technology and more.