Having read that other question, I think I understand the issue. I'll do my best without LaTeX, but this might not be pretty.
Suppose we consider everyone's utility to be of equal value. The total utility of a population would be the sum of each person's utility, i.e.
utility_total = utility_1 + utility_2 + ...
By the definition of arithmetic mean, we can see that
utility_total = mean * number_of_people. So an equivalent definition of
utility_total deals with the arithmetic mean.
Here's where I think the confusion happens:
utility_total does not deal only with the mean. It also includes the number of people.
So if adding this person would decrease the mean utility, that would be OK so long as net utility was increased. If their utility was positive, then it is guaranteed to increase net utility, so essentially the question "should we bring person X into the world" is just "will person X have positive utility?" Which (in your scenario) is independent of others' utilities, and so there is no problem.
EDIT: to be clear, this is what is sometimes known as "total utilitarianism". There does exist a variant of utilitarianism which does consider solely the average utility. In its naive form, your criticism is valid. I think proponents usually have some sort of "two-level" thinking whereby they switch between total and average utilitarianism as necessary.
To the best of my knowledge, "utilitarianism" usually means "total utilitarianism", which is why I answered this question this way. If you were interested in a defense of average utilitarianism, let me know.
EDIT 2: There are many possible solutions to the repugnant conclusion. So I'll give my favorite: there is no such thing as a life worth living.
I like it partly for shock value, but it also makes a good point: there is a fundamental difference between "this life is so good it requires you to be born" and "this life is so good it requires you to not die." I.e. once you've been born, moral laws apply to you which didn't apply before you were born.