Age of the universe versus absolute time In Wikipedia, the age of the universe is defined as the "time elapsed since the Big Bang" while "time" links to "the cosmological time parameter of comoving coordinates" which itself links to "the elapsed time since the Big Bang according to a clock of a comoving observer", the latter being defined as "the only observer that will perceive the universe, including the cosmic microwave background radiation, to be isotropic".
Meanwhile, we also find in Wikipedia: "The theory of relativity does not allow the existence of absolute time because of the nonexistence of absolute simultaneity. Absolute simultaneity refers to the experimental establishment of coincidence of two or more events in time at different locations in space in a manner agreed upon by all observers in the universe."
While both makes sense to me, I feel like a contradiction between them in the sense that the age of the universe for the comoving observer located where and when an event occurs could be considered as the absolute time at which this event occurs. Indeed, such a definition of time would probably be impossible to implement in practice due to measurement uncertainties. However, it could be used in principle to define in which order two events actually occur in a way on which all possible observers should agree.
So, what did I miss and how can we reconcile these two points of view?
 A: A comoving observer and an observer that has been moving at $0.866c$ since Big Bang will disagree on their measured age of the Universe by a factor of 2. While both measurements are correct, we can say that the comoving observer measures a more "natural" age of the Universe. For instance, the comoving observer is the only observer who will measure the Universe to be isotropic. But a time being more natural, or making more sense in terms of calculations, does not make it "absolute".
It's true that we can use comoving coordinates to define which event happened first, $A$ or $B$. And I agree that these coordinates make sense to define as "natural" coordinates. But you're free to use any other set of coordinates, and no-one is allowed you call your coordinates, or your measurements, "wrong". If you travel at $v=.866c$ toward Betelgeuze, and you observe $A$ and $B$ to happen simultaneously, this is reality$^\mathrm{\tiny{TM}}$ in your reference frame; it's not an optical illusion.
See also this similar question on the diameter of the Universe.
A: Suppose two observers, Alice and Bob, are moving relative to each other since the beginning of the universe. While they do it, they construct the chronologies of all the events of the universe, as they record them in their frame of reference.
They will construct different chronologies.
However, and this is key, each can reconstruct the other's chronology. This is also the content of special relativity. If Alice takes all her data of spacetime events, and knows Bob's velocity, she can reconstruct Bob's chronology.  Alice will always agree with Bob that event $x$ happens simultaneously with event $y$ in Bob's frame. A third party, Charlie, will also be able to agree with Alice about the order of events, as recorded by Bob.
By this, I am illustrating that, even in special relativity, all observers can agree on the order of events, as they happen in a particular frame of reference. Being able to agree about the order of events in a particular frame does not mean that you have found an absolute time, it just means that relativity is consistent!
The same is true for the comoving frame.
In our standard model of the universe, there happens to be one frame of reference which is at rest with respect to the CMB. In principle, no matter where you are in the universe, you can determine if you are at rest with respect to the CMB and thus with this particular frame (if you are, you are called a comoving observer). It is to simplify calculations that we use the comoving time as the measure of the age of the universe. It is also useful because the Earth is approximately comoving.
Being able to agree in what order events happen in the comoving frame does not mean that the comoving frame define an absolute time, it just means that you know how fast you have been moving with respect to the comoving frame, and that you know how to calculate.
A: Pulling together what's been said in various comments:
1)  General relativity admits models where spacetime is foliated by spacelike leaves, all of which are indexed by a global time coordinate.  The simplest of these models is Minkowski space.  All of your observations about models with comoving observers apply equally well to Minkowski space, so if you want to clarify the source of your confusion, that's the example you should focus on.
2)  In Minkowski Space, it's easy to define a single global time coordinate.  That doesn't contradict relativity, becauses it doesn't deny that there are also other ways to define coordinates.
3)  Models with comoving observers assume isotropy, which is clearly a considerable abstraction from reality.  Such models are useful for some purposes and less useful for others.   The existence of such models does not deny the existence of other models without global time coordinates.
4)  So the existence of a global time coordinate does not contradict relativity for two reasons:  First, as in Minkowski space, the existence of this coordinate does not deny the existence of alternative coordinates within the model.  Second, it does not deny the existence of other, more precise models.
