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You’re not accounting for the self-fulfilling prophecy effect. It’s not just that tall men with deep voices are unconsciously taken more seriously and regarded as leaders; it’s also that, as these men are growing up, they notice that and begin to regard themselves as leaders in turn. 
So, as a confounding factor for your two statistical arguments, which only can illustrate data trends, not explain them, one could add: short men with higher-pitched voices are less likely to be leaders because they have been less likely to be regarded as leaders by others and so they don’t come to regard themselves as that as frequently as tall, lower-pitched voice men do. 
If you account for the feedback between self-regard and other-regard, you can’t just unambiguously separate “quality” from bias; it turns out that people who fit the natural bias are those who come to possess the qualities in question.
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