Yes, I think there's a lot more room for personal responsibility in America.
It's definitely the case that people sometimes say "Well, they just need more personal responsibility" as a way to write-off challenges by other (often poor, often black/Hispanic) people without removing the plank in their own eye. (See
Just World Hypothesis.)
But it's our moral responsibility to help people make good decisions just as much as it is our job to keep them from going broke. I definitely used to be a believer that we just need more taxes on the wealthy to erase all the inequality in society. But after getting out in the world and seeing friends from rougher backgrounds make terrible financial decisions or get into drug problems, I realize how hard it is do well in the world when you're not surrounded by a culture/environment that supports good decision-making.
Don't get me wrong, I fully support things like progressive taxation and unemployment insurance. But government policies don't and can't go everywhere, so we need to make our society equips our citizens with the ability to make good decisions when they encounter a situation that doesn't have a specific government regulation applying to it. When we don't teach people these things, the result is that only people who grow up in stable, well-educated families have access to financial or medical decision-making skills, and that's another form of inequality.
If we can teach French history in school, surely we could make sure students get a course on personal finance--why don't we? If we want the wealthy to pay their fair share instead of moving money into tax shelters, we need to make duty to society/government an American value again--how about we get rid of Howard Zinn and bring back the Pledge of Allegiance? If we're worried about sexually-transmitted diseases and kids growing up in single-parent homes, why are allowing so much smut on the radio and TV instead of emphasizing romantic/sexual commitment?
Our focus on individualism and instant gratification is really a problem at all our levels of society. There's a direct relation between telling to "be themselves" at the expense of everyone else and hedge fund managers risking our entire economy to advance their personal wealth, between schools only teaching everything bad about American history and government and libertarian nutjobs who won't let the government vaccinate their kids.