In any spacetime you care to do an ADM split on, there are an infinite number of real-valued smooth scalar fields whose gradient is everywhere non-zero and timelike available to serve as the coordinate time.
In the standard cosmology the at-rest isotropic and homogeneous distribution of matter provides an obvious coordinate time function, but physics still has to work for other inertial and accelerated observers, so there is nothing preventing anyone from using a non-comoving observer's proper time in the ADM split.
Nits:
* Hamiltonian formulation of general relativity
* ADM formalism
The expansion is in the metric, and most visible (and most amenable to interpretation as an expansion of space) when using comoving coordinates. However, we are allowed to work in any system of coordinates, and when we do a general coordinate transform, we lose the interpretation of the metric expansion as an expansion in space.
("We see in both [Milne & de Sitter] cosmologies that the interpretation of redshift as an expansion of space is dependent upon the coordinates one chooses to calculate z." -- https://academic.oup.com/mnras/article/422/2/1418/1036317 final sentence in §3.4 de Sitter space).
The standard cosmology's FLRW spacetime (and also de Sitter space, where the vacuum constant positive scalar curvature makes for an easier-to-understand expansion history) is time-orientable. There are more points in spacetime in the future direction from an arbitrary point anywhere in the entire spacetime, than in that point's past.
However, the real distribution of matter is in our universe lumpy, as is easily demonstrated right here on Earth. Standing on the ground you can tell that locally there is neither isotropy nor homogenity (you can see quite far up into the sky, but not so far into the ground). Moreover, there are differences in isotropic pressure, convection, heat conduction, and shear stresses between the air and ground, none of which are features of the perfect fluids in the FLRW model. So locally the FLRW metric is not suitable, and since the Earth is very much not vacuum de Sitter is even less suitable. Consequently it should not offend anyone to read that Manhattan isn't undergoing cosmological expansion. Most things within ten kiloparsecs of here are gravitationally collapsing. It requires coordinate contortions to interpret interplanetary or interstellar space in the Milky way as expanding, and switching to different systems of coordinates will blow up that interpretation.
A "swiss cheese" universe with collapsing vacuoles in an expanding cosmology reflects that it's only when we go from ~kpc length scales to ~Gpc length scales that the distribution of matter behaves enough like perfect fluids to allow for an exact solution to the Friedmann equations. The collapsing vacuoles are usually modelled like Lemaître-Tolman-Bondi with a thin shell boundary; this results in a vacuole that time-orientable and has fewer spacetime points in the future direction. "Hole" time functions and "cheese" time functions must be different (hole clocks tick slower as they fall towards the centre).