Why the red-shift of distant galaxies is considered to be the effect of expanding spacetime?

If space were non-expanding the redshifts could indeed just be described as a (relativistic) red shift, but ...

... if space were non-expanding the universe would have to be a very strange place. For a start General Relativity would be disproved (because GR predicts space must be expanding or contracting) but GR passes lots of experimental tests so we're fairly confident it's correct. You'd also have to explain how the Cosmic Microwave Background exists if there wasn't a Big Bang. Finally of course you'd have to explain how come Hubble's Law holds.

So the reason why we say the redshift is due to the expansion of space is because there is so much evidence that the universe is expanding, and the expansion naturally explains the red shifts.

You might be interested in skimming the Expanding Confusion paper by Davis and Lineweaver, which I've referenced many times on this site. In particular, the last sentence of the abstract states:

We analyze apparent magnitudes of supernovae and observationally rule out the special relativistic Doppler interpretation of cosmological redshifts at a confidence level of 23$\sigma$.

The basic idea is looking at the luminosity distance using Type Ia supernovae (the "standard candles" of cosmology). The results you'd predict from a Doppler shift - even a special relativistic one - do not agree with reality. You'd have to have some pretty ad hoc additions to your special relativistic cosmology to get these results. Section 4.2 of the paper covers this.

It's not surprising, though, that there is so much confusion. The authors themselves got confused, and an earlier version of Section 4.1 stated that you could differentiate between Doppler shifts and GR based on timing with standard clocks, but in fact the two effects are indistinguishable. Perhaps the best part of the paper is Appendix B, which gives some 25 examples of misleading or outright erroneous statements on this and related topics made by well-known physicists.

I also should add that we also have the CMB radiation to observe, and it provides a unique local at-rest reference frame. That is, if you are moving with respect to the CMB, you'll know because it will appear hotter in the forward direction and cooler in the opposite direction. Now we have a small velocity relative to it, but nothing close to the speed of light. So if you try to explain redshifts with just SR and the Doppler effect, you have to explain why we happen to live in one of the 100 or so galaxies barely moving with respect to the CMB, and why the billions of other galaxies we observe are all moving quite fast with respect to it. In GR, everyone can be locally "at rest" and see an isotropic CMB, while still seeing each other redshifted. So there's a Copernican argument for you - the SR description forces you to believe you are in a special location in the universe.

If you want to intuitively understand the cosmological redshift, I'll shamelessly plug my own excessively verbose answer to a related question.