This is more a comment than an answer: a few years back, in 2011, while working with some friends on SGA1, we also found out that we could not prove this statement without the hypothesis that $X$ is quasi-separated. Our question: Is this hypothesis simply missing in SGA1 ? reached Michel Raynaud and his answer was reported to be something like  : Probably, but this is not very interesting.

This is more a comment than an answer: a few years back, in 2011, while working with some friends on SGA1, we also found out that we could not prove this statement without the hypothesis that $X$ is quasi-separated. Our question: Is this hypothesis simply missing in SGA1 ? reached Michel Raynaud and his answer was reported to be something like: Probably, but this is not very interesting.

This is more a comment than an answer  : a few years back, in 2011, while working with some friends on SGA1, we also found out that we could not prove this statement without the hypothesis that $X$ is quasi-separated. Our question  : Is this hypothesis simply missing in SGA1 ? reached Michel Raynaud and his answer was reported to be something like : Probably, but this is not very interesting.

This is more a comment than an answer: a few years back, in 2011, while working with some friends on SGA1, we also found out that we could not prove this statement without the hypothesis that $X$ is quasi-separated. Our question: Is this hypothesis simply missing in SGA1 ? reached Michel Raynaud and his answer was reported to be something like : Probably, but this is not very interesting.