I don't think people will care until the issue has become large enough that it will start impacting others in society indirectly. Eg not enough tax revenue being generated or not enough men willing to fight in a war.
With the draft, people from all walks of life were enlisted. (Yes, there were ways to weasel out of it, but beside the point)
Without the draft, the army is largely built from the lower-class, conservative, patriotic populous.
The argument is that the privileged would be less likely to send people off to war if it meant their loved ones lives were on the line.
Today I’m married with a child, I support my family as the primary breadwinner, I have a BS and MBA and am pursuing a PhD in computer science. All of that education was paid for by the GI Bill.
Sample size of one, but I can confirm your hypothesis.