Was Nazi propaganda successful enough so that people believed they should be killed if old or disabled, for the benefit of the nation?
Or, was this a temporary scheme to lessen expenses on healthcare, that was not intended to be continued in the long-term?
This is a complicated topic and most of my reference library is at work, so...
First, Rassenhygiene / "racial hygiene" was something that had been developing for years even before the Nazis came into power. The idea that "Ballastexistenzen" / literally people whose existence were a burden and were threatening the gene pool and the health of the whole society was an idea that gripped many different countries. This means that the Nazis didn't bring up an unusual, unheard-of idea when Nazi "Euthanasia" started.
The economic cost of the disabled (and especially those who were seen as mentally ill) for the economy was something that was always factoring into this as well. If you check out Nazi propaganda, economic impact and "Volksgesundheit" were both similarly connected to the idea of "racial hygiene".
Now, "Euthanasie" / Nazi euthanasia was supposed to be "sold" to Germans based on ending the suffering of those who were incureable. The so-called "T4-Aktion" that was implemented in the end, though, was secretly enacted, centrally organized and was supposed to operate as low-key as possible. Hitler (rightly, as it turned out) feared the impact of rumors about "mercy killings". While it is still very much open for debate what the deciding factor was for ending the "T4"-killings, many researchers cite the speech by Bishop von Galen (Munster) who openly talked about the murders and / or the upcoming war in the east and the necessity to keep up morale in the German population.
This means that while Nazi propaganda had a good baseline to work with and was well thought-out, Hitler still didn't seem to trust that there would be any kind of consensus about the necessity of killing the sick and disabled.
Generalizing about the opinions of "the people" is hard. I can only tell you what I heard from people myself. I work in a memorial museums depicting the "T4"-killings and I've personally met people who say that having a "T4"-victim in the family was a constant source of shame for the family. And, in some cases, is, until today.
Now, the expanding circle of possible victims of Nazi "Euthanasia" - from "disabled" people to traumatized citizens and even some wounded soldiers (and lets not forget the Nazi slave laborers) - was not part of the "T4"-project. Many still call it "wild" "Euthanasia", but I agree with my colleagues that "regionalised" is the better word. Local bureaucrats and institution directors were acting on their own accord, and it is hard to generalize what happened. That someone who was traumatized by heavy bombing was brought into one of the "Euthanasia"-instiutions and killed is noteworthy, but it is extremely hard to say if the "typical" German who was exposed to a lot of propaganda would've even connected this killing to the concept of "mercy killings".
If you want to know more, "Benzenhöfer, Udo; Radosh, Anthony B.: Euthanasia in Germany. Before and during the Third Reich, Münster 2010" is not a bad place to start. I'm not a big fan of Friedlanders book, because the direct connection established between Holocaust and "T4" he is so clearly stating is not something I would 100% support, but that's open for debate. The post-war trials are also very interesting from a historical standpoint, if you're interested you could take a look into "Bryant, Michael S.: Confronting the "good death. Nazi euthanasia on trial, 1945-1953, Boulder, CO 2005". Also, I've heard good things about "Gehring-Münzel, Ursula; Dallas, Marci Regan; Perry, Ira D.; Stein, Renata (Hgg.): How healing becomes killing. Eugenics, euthanasia, extermination, Houston, TX 2007".
I hope this helps even with less footnotes than I would usually add.