Until this latest economic crisis, the 'average' American had a full time job, and therefore, benefits. Being part-time with limited income was something that happened to 'other people'. Now, people are working two and three part time jobs - if they can find any - just to make bills and house/car payments. Therefore, it's now something that the 'average American' suddenly has an interest in.
I think the average American adult still has a full time job, underemployment is only something like, 22%. Your point stands though, even if people aren't experiencing it personally they know someone who is, and by extension they've come face to face with this reality.
I'd say the people who have healthcare are still, by and large, the people who "don't see a problem" with the current system, and I can understand why. Right now things are being rationed according to wealth, but even in a non-profit system rationing is still going to be necessary, how and why are just going to be a different conversation. When it's done according to wealth there's a certain degree of responsibility that falls onto the person who is "being rationed" (you don't have a job, you haven't kept your finances well, etc., that's why you're being denied healthcare).
I we were to move to a non-profit system we have to start deciding who does and doesn't get care on broader factors that may or may not be related to anything the individual is has gone. The responsibility of deciding who lives and who dies will fall on society, and that discussion will bring us face to face with a very unsavory reality: no matter what we do, we cannot give healthcare to everyone in all circumstances, so the cries of some people will always be ignored.
It's a big ugly can of worms that opposing reform helps us avoid, I kind of think that's partially responsible for the status quo.