> Does the idea of a single cosmological time even make sense?
Sure, it's just a convenient choice of coordinates. Even in a model that is not exactly homogeneous and isotropic, it can still be a convenient choice of coordinates to have cosmological time track some kind of average of the low mass and high mass regions. As I understand it, that's basically what the alternate models described in the article are doing.
> I thought one of the key parts of relativity is that which events happen simultaneously depends on your perspective.
That's true, but it doesn't actually mean very much. In a particular spacetime geometry, you can still have particular things that are picked out physically by the properties of that specific spacetime, and one of them can be "cosmological time". In an exactly homogeneous and isotropic model, that time is picked out by the symmetries of homogeneity and isotropy. But even in a model where homogeneity and isotropy are only average properties, their average is still picked out physically--for example, by the CMB, which is much, much closer to being exactly homogeneous and isotropic than galaxies and galaxy clusters (it's homogeneous and isotropic to within about 1 part in 100,000). So picking "cosmological time" based on observers who see the CMB that way is a physical method of picking them out; it's not an arbitrary choice.
Bear in mind that in special relativity, when you talk about relativity of simultaneity, you are talking about spacetime that is empty--there are no gravitating bodies anywhere. So all inertial frames are indeed equivalent in that spacetime, not just mathematically but physically. But as soon as you put gravitating bodies in, that symmetry is broken: the gravitating bodies have a definite state of motion, and that picks out certain choices of coordinates as being aligned with the gravitating bodies. So while it's true that you don't have to use those coordinates, they are convenient and they do reflect an actual physical property of the spacetime, and so does the definition of time they give.
Sure, it's just a convenient choice of coordinates. Even in a model that is not exactly homogeneous and isotropic, it can still be a convenient choice of coordinates to have cosmological time track some kind of average of the low mass and high mass regions. As I understand it, that's basically what the alternate models described in the article are doing.
> I thought one of the key parts of relativity is that which events happen simultaneously depends on your perspective.
That's true, but it doesn't actually mean very much. In a particular spacetime geometry, you can still have particular things that are picked out physically by the properties of that specific spacetime, and one of them can be "cosmological time". In an exactly homogeneous and isotropic model, that time is picked out by the symmetries of homogeneity and isotropy. But even in a model where homogeneity and isotropy are only average properties, their average is still picked out physically--for example, by the CMB, which is much, much closer to being exactly homogeneous and isotropic than galaxies and galaxy clusters (it's homogeneous and isotropic to within about 1 part in 100,000). So picking "cosmological time" based on observers who see the CMB that way is a physical method of picking them out; it's not an arbitrary choice.
Bear in mind that in special relativity, when you talk about relativity of simultaneity, you are talking about spacetime that is empty--there are no gravitating bodies anywhere. So all inertial frames are indeed equivalent in that spacetime, not just mathematically but physically. But as soon as you put gravitating bodies in, that symmetry is broken: the gravitating bodies have a definite state of motion, and that picks out certain choices of coordinates as being aligned with the gravitating bodies. So while it's true that you don't have to use those coordinates, they are convenient and they do reflect an actual physical property of the spacetime, and so does the definition of time they give.