My explanations could be dumbed down I guess. You mean stuff like this:
So I jumped over your head. I simply did not say the excited elements give off the photons. This is a given for anyone understanding spectral theory. I guess you don't.
Various things can cause redshift. Moving through a varying time, velocity changes, gravity, changes in media density, etc. In space, we don't expect density changes and rarely gravity or time changes. Two moving objects will have a vector relationship where you have to apply a variation of vector mathematics, though the observer will still move as the light does.
The bottom line is there are several causes for red shifting, but the Doppler effect is still part of it when there is relative moment.