If light speed determines the rate of time, could the speed of light be varying over time and we be unaware of it? Is this a valid implication/alternative explanation of universal red-shift? I thought I'd ask this when I read this question about light clocks.
I had speculated this way: If light speed determines the rate of time, could the speed of light be varying over time and we be unaware of it? We measure light speed by itself.
But if light speed was varying over a scale of minutes or hours, we'd see variation in the Sun's spectrum over time, either red shift or blue shift, and we don't.
If it was varying over a period of years, we'd see variation in the spectra of stars, and we don't.
If it was varying over a period of billions of years, we'd see variation in the spectrums of distant galaxies, and we do. 
Is this a possible implication of universal red-shift? Or even an alternative to the expansion explanation? 
 A: Probably not- here is one explanation. 
When we look at the most distant objects in the universe (super-bright galaxies or active galactic cores) we are seeing them in an early stage of their lifetimes. It is thought that at those times, the supermassive black holes in those galaxy centers were in the process of actively gobbling up dust and gas in their neighborhoods, which there (apparently) was plenty of at that time. 
This means at early times, the spectra of those high-brightness galaxies was dominated by the output of extremely hot accretion discs surrounding those black holes and huge and super-energetic jets emitted from near the poles of those black holes. 
Then, as time passed, the galaxy cores got depleted in readily-sucked-up material and the  black hole neighborhoods grew quiet. The black holes are still there, of course, but with little to feed on, the light sources they powered went out.
The spectra of galaxies without active cores then became dominated by starlight instead of by accretion processes and jets. 
A: 
If it was varying over a period of billions of years, we'd see
  variation in the spectrums of distant galaxies, and we do.
Is this a possible implication of universal red-shift? Or even an
  alternative to the expansion explanation?

This has been explored before in both scientific and creationist lines.
The problem is that we can see the spectra within the red-shifted light, and the spectra are the same as they are here, just shifted. So for instance, the relative energy levels of the hydrogen atoms are the same in distance galaxies as they are here on Earth.
That would not be true if it was the speed of light that was changing. One could likely offset some of these observations by carefully selecting countering changes to other universal constants, but it seems unlikely there is any set that would fully explain our (rather deep) observational set.
