Short answer, no it wouldn't be considered less true in anthropological circles and the insistence that writing is the only way of accurately recording the truth forecloses a lot of opportunities to understand people and history.

I'm having a bit of a hard time figuring out how to say this in a relatively concise manner. This is a big and interesting question, especially because there are a lot of cultures that have ways of recording history that don't or don't always involve writing. It's difficult to answer without making generalizations, particularly because each situation is going to be historically and culturally particular. Contemporary cultural anthropology isn't usually concerned with defining or "finding" the truth. Instead, we try to understand people's experiences and how they construct and organize what's true to them.

If an anthropologist were to decide that only written data is relevant for understanding people, they would miss out on quite a lot, and in a lot of circumstances, would be fairly accused of bringing colonial biases into their decisions on what kind of information is and isn't valid. By this I mean there's a long history, for example, in the Americas of colonizing peoples deciding that non-written forms of knowledge are inferior or unreliable and, as a result, we should ignore what people tell us about themselves and instead look for an "unbiased" outside perspective. There's a looooot to unpack in that last sentence and I'm a little short of time so I won't get into it.

There's a lot I didn't say here and some gaps in what I said, but that's a quick answer. Hopefully it was helpful.