What if it light just loses energy as it travels, so that the frequency shifts lower? That seems like a perfectly natural solution. How do we know it isn't true?
As gjm mentions, the general name for this sort of theory is "tired light." And these theories have been studied extensively and they are broken.
We have a very accurate, very well-tested theory that describes the way photons behave, quantum electrodynamics. It predicts that photons in the vacuum have a constant frequency and don't suddenly vanish. Nor do photons have any sort of internal "clock" for how long they have been propagating. As near as I can tell, any sort of tired light model means giving up QED in fairly fundamental ways, and the evidentiary bar to overturn that theory is very high.
Worse, tired light seems to break local energy conservation. If photons just vanish or spontaneously redshift, where does the energy go?
I can conceive of there being a tired light model that isn't ruled out by experiment, but I would like to see that theory before I junk all of 20th century cosmology and fundamental physics.
Most scientific theories, most of the time, have a whole bunch of quirky observations that they don't explain well. Mostly these anomalies gradually go away as people find bugs in the experiments, or take into account various effects they hadn't considered. The astronomical anomalies you point to don't seem remotely problematic enough to give up on modern physics.
I am submitting this on behalf of MazeHatter, who originally posted it here in the most recent open tread. Go there to upvote if you like this submission.
Begin MazeHatter:
I grew up thinking that the Big Bang was the beginning of it all. In 2013 and 2014 a good number of observations have thrown some of our basic assumptions about the theory into question. There were anomalies observed in the CMB, previously ignored, now confirmed by Planck:
http://www.esa.int/Our_Activities/Space_Science/Planck/Planck_reveals_an_almost_perfect_Universe
We are also getting a better look at galaxies at greater distances, thinking they would all be young galaxies, and finding they are not:
http://carnegiescience.edu/news/some_galaxies_early_universe_grew_quickly
http://mq.edu.au/newsroom/2014/03/11/granny-galaxies-discovered-in-the-early-universe/
B. D. Simmons et al. Galaxy Zoo: CANDELS Barred Disks and Bar Fractions. Monthly Notices of the Royal Astronomical Society, 2014 DOI: 10.1093/mnras/stu1817
http://www.sciencedaily.com/releases/2014/10/141030101241.htm
http://www.nasa.gov/jpl/spitzer/splash-project-dives-deep-for-galaxies/#.VBxS4o938jg
Although it seems we don't have to look so far away to find evidence that galaxy formation is inconsistent with the Big Bang timeline.
http://www.natureworldnews.com/articles/7528/20140611/galaxy-formation-theories-undermined-dwarf-galaxies.htm
http://arxiv.org/abs/1406.1799
Another observation is that lithium abundances are way too low for the theory in other places, not just here:
http://news.nationalgeographic.com/news/2014/09/140910-space-lithium-m54-star-cluster-science/
It also seems there is larger scale structure continually being discovered larger than the Big Bang is thought to account for:
http://www.sciencedaily.com/releases/2014/11/141119084506.htm
D. Hutsemékers, L. Braibant, V. Pelgrims, D. Sluse. Alignment of quasar polarizations with large-scale structures. Astronomy & Astrophysics, 2014
http://www.sciencedaily.com/releases/2013/01/130111092539.htm
These observations have been made just recently. It seems that in the 1980's, when I was first introduced to the Big Bang as a child, the experts in the field knew then there were problems with it, and devised inflation as a solution. And today, the validity of that solution is being called into question by those same experts:
http://www.physics.princeton.edu/~steinh/0411036.pdf
What are the odds 2015 will be more like 2014 where we (again) found larger and older galaxies at greater distances, or will it be more like 1983?