You're right, there is an extra layer of conditionals. We need to incorporate the information about what was said ("One of the kids is a girl") into the calculations as well.

Spoiler for elaboration

The equation that you wrote below in green is correct mathematically. (I replaced your 'If' with 'Given' since it fits better into the Bayesian framework)

p(Other is girl) =

p(Other is girl Given Ned spoke) x p(Ned spoke) +

p(Other is girl Given Red spoke) x p(Red spoke) +

p(Other is girl Given Ted spoke) x p(Ted spoke) +

p(Other is girl Given Zed spoke) x p(Zed spoke).

However, the left hand side, P(Other is a girl), is not the solution to this puzzle. The probability that we want is the probability that "Other is a girl" given that a reporter (we don't know which one) said 'One kid is a girl'. That is, we want P( 'Other is a girl' Given 'One kid is a girl').

From my derivations in post #14, we can see that P( 'Other is a girl' Given 'One kid is a girl') can be expressed as

P('Other is a girl' Given 'One kid is a girl' ) =

p('Other is girl' Given [ Ned spoke and 'One kid is a girl' ]) x p(Ned spoke Given 'One kid is a girl')

p('Other is girl' Given [ Red spoke and 'One kid is a girl' ]) x p(Red spoke Given 'One kid is a girl')

p('Other is girl' Given [ Ted spoke and 'One kid is a girl' ]) x p(Ted spoke Given 'One kid is a girl')

p('Other is girl' Given [ Zed spoke and 'One kid is a girl' ]) x p(Zed spoke Given 'One kid is a girl').

Essentially, we don't know which reporter is the speaker, so take a weighted average over all reporters based on the conditional probabilities P( Reporter Given 'One kid is a girl'). Now let's examine values of these probabilities

Since uncle Jed drew the names out of the hat, we have

p(Ned spoke) = p(Red spoke) = p(Ted spoke) = p(Zed spoke) = 1/4

From Yoruichi-san's analysis, we have

p(Ned spoke Given 'One kid is a girl') = 1/7

p(Red spoke Given 'One kid is a girl') = 3/7

p(Ted spoke Given 'One kid is a girl') = 2/7

p(Zed spoke Given 'One kid is a girl') = 1/7

From the equation in green above, it seems that post #10 was solving for p(Other is girl Given Ned spoke) and p(Other is girl Given Red spoke) and so on. However, those probabilities are actually different conditionals. They actually are

p('Other is girl' Given [ Ned spoke and 'One kid is a girl' ]) = 1

p('Other is girl' Given [ Red spoke and 'One kid is a girl' ]) = 1/3

p('Other is girl' Given [ Ted spoke and 'One kid is a girl' ]) = 1/2

p('Other is girl' Given [ Zed spoke and 'One kid is a girl' ]) = 1

So I believe the mistakes in post #10 are as follows

1) It solves for a different quantity, P(Other is a girl) versus P(Other is a girl Given 'One kid is a girl'), with respect to the solution

2) It actually computed p('Other is girl' Given [ Ned spoke and 'One kid is a girl' ]) instead of p('Other is girl' Given Ned spoke) and so on.

Thanks. That makes sense now.