With the recent advancement of Large Language Models (LLMs), efforts have been made to leverage LLMs in crucial social science study methods, including predicting human features of social life such as presidential voting. Existing works suggest that LLMs are capable of generating human-like responses. Nevertheless, it is unclear how well LLMs work and where the plausible predictions derive from. This paper critically examines the performance of LLMs as social predictors, pointing out the source of correct predictions and limitations. Based on the notion of mutability that classifies social features, we design three realistic settings and a novel social prediction task, where the LLMs make predictions with input features of the same mutability and accessibility with the response feature. We find that the promising performance achieved by previous studies is because of input shortcut features to the response, which are hard to capture in reality; the performance degrades dramatically to near-random after removing the shortcuts. With the comprehensive investigations on various LLMs, we reveal that LLMs struggle to work as expected on social prediction when given ordinarily available input features without shortcuts. We further investigate possible reasons for this phenomenon and suggest potential ways to enhance LLMs for social prediction.