Artificial Intelligence applications show great potential for language-related tasks that rely on next-word prediction. The current generation of large language models have been linked to claims about human-like linguistic performance and their applications are hailed both as a key step towards Artificial General Intelligence and as major advance in understanding the cognitive, and even neural basis of human language. We analyze the contribution of large language models as theoretically informative representations of a target system vs. atheoretical powerful mechanistic tools, and we identify the key abilities that are still missing from the current state of development and exploitation of these models.