OK, let us try what you suggest.

In the standard FLRW (Friedmann-Lemaitre-Robertson-Walker) cosmology, it is assumed that space is homogeneous and isotropic. As a result, we can write the metric in the form [math]ds^2=dt^2-a(t)^2dR^2[/math] (where [math]dR^2=dx^2+dy^2+dz^2[/math] or however else you wish to represent space.)

Spatial distances in this cosmology will be scaled by [math]a(t)[/math], a function of time, and the rate of expansion is represented by [math]H(t)=\dot{a}(t)/a(t)[/math] (overdot representing differentiation with respect to time), the value of which, with units of inverse time, is the Hubble "constant" at the present epoch.

But the FLRW metric can be trivially rewritten using new spatial coordinates [math]R'=a^{-1}R,[/math] which gives*

[math]\begin{align} R'&=aR,\\ dR&=d(a^{-1}R')=-a^{-2}\dot{a}R'dt+a^{-1}dR',\\ ds^2&=dt^2-a^2[-a^{-2}\dot{a}R'dt+a^{-1}dR']^2\\ &=(1-a^{-2}\dot{a}^2{R'}^2)dt^2+2a^{-1}\dot{a}R'dtdR'-d{R'}^2\\ &=(1-H^2{R'}^2)dt^2+2HR'dtdR'-d{R'}^2. \end{align}\tag*{}[/math]

I.e., we just reinterpreted a universe of expanding space as a universe of shrinking time. The rate of shrinking is governed by [math]H,[/math] which is the same quantity (expressed in units of inverse time) as before, and its present value is still Hubble's "constant".

These two pictures, which are connected by a trivial mathematical transformation, are of course equivalent. And a basic tenet of general relativity is that the physics should be independent of the choice of coordinate system, so whether we use [math](t,R)[/math] or [math](t,R')[/math] as our coordinates should be irrelevant.

Moreover, if we were to derive the equations that govern the evolution of spacetime (the Friedmann equations), in both cases the same set of equations will result: two equations that relate the scale factor [math]a(t)[/math] as a function of time to the density and pressure (also functions of time) of the universe. If a third function that relates density to pressure (the equation of state) is also given, we can solve the system and make assertions about the universe as it evolves.

So is there any real difference between interpreting the Hubble redshift as an expansion of space or as a shrinking of the unit of time? I'd argue that there isn't. In both pictures, we predict that as time passes, it will take longer for a ray of light to cross the distance between two distant galaxies. Whether it is because the distance between them is increasing or because the unit of time shrank makes no difference when it comes to observation. Similarly, we would predict that the longer a ray of light traveled before it reached our instruments, the more redshifted it will be. Again, whether because it is due to a distance-dependent velocity of recession or an age-dependent change in the unit of time is irrelevant: the observational prediction is the same in both cases.

Lastly, in both cases it is possible for small deviations from homogeneity to emerge and ensure that no expansion (or no changes in the unit of time) happen in bound systems like inside a galaxy cluster, a galaxy, or a star or planet.

There is, however, one point in favor of treating the observation as a spatial expansion. That is, when you look at the expansion in detail, it turns out not exactly isotropic after all. When that happens, we end up with a metric such as [math]ds^2=dt^2-a_x(t)^2dx^2-a_y(t)^2dy^2-a_z(t)^2dz^2,[/math] and these three independent time-dependent coefficients cannot all be absorbed into a single coefficient in front of [math]dt^2.[/math]

*Edit (May 11, 2020): I wrote this answer more close to six years ago and then forgot all about it, otherwise I’d have corrected the line element in the new metric eons ago. The correct form is not essential to the discussion, but it was still incorrect, with missing terms and the wrong sign in one place.