Then again, Weird, anxiety is also fairly common in children who are not gifted at all.
Out of my 3 siblings, the one who worried most was the one distinctly average academically. Had to work hard to get Bs and Cs, but worried about everything from Russian submarines to the family running out of food (parents in steady employment, middle class jobs, was never going to happen).
I was possibly more of a gifted child, at least in some areas. But I've always had a tendency to plod through life cheerfully on the assumption that everything's going to work out ok.
My ds otoh, while not having any identifiable special needs, was at the bottom of his class throughout primary, but again definitely a worrier. Obsessively interested in world affairs until he got to puberty and then switched off entirely, I think because he found it too stressful.
You don't need to be very bright at all to worry about mortality or climate change aged 5: even not-very-gifted children find out about these things at about that age and if you are naturally anxious, you're going to worry.
Looking back at my extended family, what I can see is something similar to what sirfredgeorge suggests: distinct strands that combine in different ways in different individuals.
There is, in our family (not counting my adopted db who has a brand all his own), three different kinds of giftedness: one is verbal/associative, one is mathematical/analytical, the third is musical. Some individuals have one of these, some have two, some have none and one person at least (my db) combines all three.
There is also a trait of anxiety/depression which can be traced back for at least 4 generations. This is inherited by roughly half the family, but not in a way that correlates with giftedness: some are gifted and have it, some are gifted and don't have it, some are not gifted but have it etc.
It's like my grandmother's distinct nose: inherited by some but not by others, quite regardless of whether they have also inherited her short stature. Or the joint disorder which some of us have also inherited.
On the other hand, there is no autism or ADHD in the family anywhere and even my highly gifted db shows no trace of it.