"organizations" like FAIR, CIS, NumbersUSA have very very effectively controlled it/changed it. (Please Google John Tanton immigration, he's the founder or co-founder of all those groups).
These are hate groups according to SPLC (Southern Policy Law Center). These organizations have become mainstream. You can't read an immigration related article without a quote from someone from these hate groups. They come up with BS data that unfortunately gets picked up and used by lazy journalists. They coined the term "chain migration" which is a shitty way of saying family reunification. I could go on and on.
Going back to the narrative you mentioned. The gop and the dump administration have very successfully kept moving the goal posts. Before, immigration reform meant comprehensive immigration reform as in some form of amnesty for all undocumented, like the Gang of Eight bill. Now it's simply Dreamers relief plus a wall, and much tougher immigration rules and less legal immigration.
I agree with your last sentence 100%.