Yes, but so what? The colour we see is the colour we see and that seems like a sensible colour to report.

Think of it this way: If you take a picture from a mountain, things in the distance will look bluer than they are when you are close, but does this mean that all the pictures taken from mountaintops have "fake" colours that need to be corrected?

In this sense redshift is just another thing that makes things change colour as the distance increases.