Why can't we directly observe homogeneity because we "observe down the past light cone"? Homogeneity is the idea that the universe looks roughly the same no matter where the observer is. This paper on whether the universe is homogeneous includes this line:

A common misconception is that ‘homogeneity is obvious from the cosmic microwave background (CMB) and the galaxy distribution’. In fact, we cannot directly observe or test homogeneity—since we observe down the past light cone, and not on spatial surfaces that intersect that light cone (figure 1).

It comes with this figure:

Which has the caption: Figure 1. We observe down the past light cone and therefore cannot directly confirm homogeneity. (Online version in colour.)
I don't understand this argument. My intuitive understanding for why we can't directly observe homogeneity is simply that we're on Earth and can't just fly a telescope to e.g. Andromeda to see what the universe looks like from another galaxy. The argument in the paper seems to be saying that we cannot directly observe homogeneity because everything we see is in the past; however it still seems to me that if the universe was homogeneous in the past then we would expect it to be homogeneous today. Alternatively, we could in principle measure the positions & velocities of all the other galaxies in the past, evolve them forward using GR, and therefore tell if our universe is homogeneous today.
Can someone explain how the paper's argument works?
 A: This is a very good question. I hope I can explain it clearly to you here. I'll try to be pedagogical. 
Consider a tabletop spread of seeds. You can say that the distribution of seeds is homogogeneous at some length scale $L$ if the number of seeds per area $L^2$ (or unit volume $L^3$) is constant throughout the spread. 
You are allowed to make this statement because you can see the entire system from a global perspective at any instance of time. Furthermore the dynamics of this small arrangement is for this purpose instantaneous. 
In cosmology, this "Newtonian" notion of instantaneity is abolished. (ignoring quantum effects) This means that no two events where one influences the other are ever spacelike seperated and therefore never on the same spacelike hypersurface. This is simple the causal nature of spacetime. (Note that the effect of time evolution in GR is basically acting a time evolution operator on every spacelike hypersurface, leading to a  foliation of spacetime.)
Given this knowledge: What this means is that if you are observing any other system in the universe, it necessarily exists on a different spatial hypersurface and in the past light cone. 
Coming back to our analogy of seeds on a table: Now if you try to define homogeneity here, it is like trying to compare the distribution of seeds with some population density $P_A$ on table A with the distribution of seeds with some other population density $P_B$ on some other table B. It doesn't make sense anymore. 
What does alleviate this problem is isotropy. Isotropy of the universe basically states that if you look at any direction in the sky i.e. at any direction in the past light cone, changing the direction doesn't really change the observation. That means that all the points $d_1$ light years away look similar, all the points $d_2$ light years away look similar and so on and so forth. This allows us to extrapolate and make the claim that if we were capable of viewing the universe as a whole, by this logic, it should be homogeneous. 
Bottom line is that you need isotropy for homogeneity. We can't observe homogeneity, we can only infer it through isotropy.
