I'm pretty sure the leader traits remain the same with random personalities. It's more about the behavior of a leader based on their coding for various factors. So yeah, Gandhi would basically have Shaka's personality (code). It affects all kinds of things within the game such as attitudes, warprobs. wonderprob, unitprob, etc. It also changes favorite Religion and Civic. In fact, the fave civic will be a mystery until...I think...you either reach that tech or adopt that civic.
Could well be that the two traits stay with the leader name. I was only summarizing my impressions from discussions on the forums, having never used that option myself.
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