Think about it.

As a government there is no point to investing in a market that every "mere mortal" has access to. Why pay the premium for this crowdedness of the demand for it?

It makes much more sense for them to invest in things that they can EXCLUSIVELY buy. Among all the assets that governments can secure for themselves, there's a whole range of things that share the wealth preserving properties of gold without being available to the public at large.

Look at this picture; I can't guarantee that all the figures in it are accurate, but it's just there to illustrate a point:

What makes gold so special? Why is it so much more suitable as a wealth preserving asset than any of the things on that list?

Why would a government preserve its wealth in the form of gold when there are tons of things that are either illegal or impracticable for non-government parties to hold? Why would governments pay more for stores of wealth than they strictly need to by bidding on a public market rather than a tightly exclusive one?