Appealing to their moral side seems... perhaps necessary, because it seems a vocal minority simply do not want multi-family housing in their neighborhood at all. Look at the pushback by NIMBYs at city meetings across the US when anything like somewhat dense housing is proposed: right off the bat, I have literally never heard of any community collectively saying, "this sounds reasonable." I would be happy to be proven wrong.
Instead, it's pushback after pushback, claiming everything from character of the neighborhood to shadows from a tall building (even if the building is only 5 stories high, and most buildings in the neighborhood are 3 stories tall).
There's also conspicuously rare talk from those NIMBYs claiming what they do want. Instead, at the start of a project, it's always vague, "well not THAT many units!" or "well the traffic will get SO much worse!"
I've never seen specifics like, "We need 30 units or less in this proposal because of reason X and Y." Instead, it's just negotiation trying to get it as low as possible. Basically, trying to pull up the ladder as much as possible to minimize people moving to the area to folks who can afford a fairly expensive single family home.
Any single family home is fairly expensive now it seems these days, across the USA, relatively to the area it's in.
It's depressing, and I'm not sure how to get people to change those attitudes.
One thought: have people attend these meetings who are not yet residents of the neighborhoods, but would consider it if they could move into one of these developments. Of course, NIMBYs would likely be outraged that folks from outside of their neighborhood are levying their opinion... even though the NIMBYs themselves are not vocalizing considering the opinions of people who want to move to the area.