Comment by adrian_b

The most important reason is that the frequencies of optical clocks can be very different, from ultraviolet to infrared.

When the frequency of one clock is 10 times greater than the frequency of another, it is hard to find a significance for any other kind of mean, except the geometric.

More clearly, if the input frequencies have the same relative uncertainty, the geometric mean will preserve that relative uncertainty. If the input frequencies have different relative uncertainties, the geometric mean will have a relative uncertainty that is intermediate between them.

Other kinds of means do not offer guarantees about the relative uncertainty when the ratio of the inputs is high. If one frequency is much bigger than the others, the arithmetic mean will depend only on that frequency, the others will not matter.