The two most obvious solutions to the thought experiment presented are either 1) space is absolute in some way (i.e. the classical Newtonian response) or 2) the behavior of space "here" is affected by the by distribution of matter "over there". General relativity gives us a strong argument in favor of (2) by showing that a) many physical principles thought to be absolute are actually relative and b) showing that mass "over there" affects the shape of space "here".

To say anything more concrete requires requires defining the question much more precisely. I believe there is still some disagreement on the interpretation of Mach's principle in light of general relativity. For example, see https://en.wikipedia.org/wiki/Mach's_principle#Variations_in... (and a couple sections above, the 1993 poll of physicists asking: "Is general relativity with appropriate boundary conditions of closure of some kind very Machian?"

I hope that is helpful in some way.

The unsatisfying mathematical answer is that it is impossible to have a uniform distribution of rotational speeds, therefore there must be a preferred one.

It's the same reason the universe has an average speed (unlike what you might expect from special relativity), although it is unclear if this is true for the entire universe or just the portion we can see. We can measure how fast we're moving w.r.t the cosmic microwave background radiation though (it is red-/blue-shifted in a particular direction).

This is an interesting argument! Wouldn't it also work for positions, though? That is, either the universe is finite, or, since there can't be a uniform distribution over an infinite space of positions, there must be some preferred "center" of the universe?

You'd think, but of course we know that not to be the case. It's hard to pinpoint the exact reason though. Sure we know time and space are rather special, but its hard to say exactly why.

In the end though I reckon the most obvious reason is that speed is a property that directly corresponds to energy, therefore for each region of space to have a well defined energy (which is required for e.g. general relativity) every region of space needs to have a well defined distribution of speeds.

I suppose this does leave open a small loophole, as you can easily correlate speed with position in order to get a distribution that is uniform in both (but correlated). But this goes against our assumption that the universe is uniform everywhere (which might turn out to be false, but so far it's holding up well).