It's not "out of favor" just as a matter of fashion: a lot more data was collected, extending much further away and therefore much further into the past.
> a few more new facts just might swing things the other way again.
No, it would take a lot more than that, because our current model of accelerating expansion for the past few billion years (note that before that, the expansion was decelerating) is based on a lot more data than we had 40 or so years ago.
Afaik, we have no direct observations of galaxies moving away from us - over such enormous distances we have not had enough time since the start of observations to actually see a galaxy recede.
The change in wavelength is exactly proportional to the increase in size of the universe during the time of flight, and it is not directly related to speed of recession except in common and inaccurate discussion. The link to relative velocity (if that concept makes sense for objects in widely separated and unrelated reference frames) is model dependent, while the metric expansion measurement is directly measured.