The answer to this is, of course, a matter of personal opinion as there is no way to objectively say what is and is not a “key” part of our political culture. My own opinion is that anyone who says that individual responsibility is no longer a key part of our culture is vastly overstating their case.
I would like to address this by making the following points”
We still believe strongly in personal responsibility.
We overrate the degree to which we believed in personal responsibility in the past.
We cannot possibly think that we can have a modern society in which people are completely responsible for their own lives.
First, there is a huge difference between saying that we have a welfare state and saying that we no longer believe in personal responsibility. Let us look at the first three aspects of the welfare state that you mention here. You talk about unemployment insurance, pensions, and educational subsidies. I would argue that all of these things are actually compatible with personal responsibility. These are not things the government just gives us. We have to work to earn the right to collect unemployment insurance. We have to contribute to Social Security, and even that is not enough to ensure a good standard of living for most people. When we get educational subsidies, we often have to repay them (since they come in the form of loans) and we are personally responsible for getting and using our education.
Second, this question implies that we were once completely self-sufficient in the past but are no longer so. This is simply not true. We think the “west was won” without government help. In reality, the government subsidized the railroads that brought the settlers to the West. The government provided the military that pushed the Native Americans away from the land that the white settlers wanted. The government provided the land that the farmers homesteaded and government land grant colleges helped the farmers learn how to farm more efficiently. All of these are examples of how we depended on the government in the past just as we do now.
Finally, it simply is not possible to have a modern society in which everyone is completely autonomous. In our modern society, we are all interconnected because we are not simply small farmers who make or grow all the things we need. If we do not help the poor have housing and medical care, we end up with homeless people who harm our society. We end up with children who will have a hard time getting a good education because their parents are poor, homeless, sick, or all three. These things hurt our society because we cannot simply ignore these people. They might end up being imprisoned and costing us money. They might end up being poorly educated and therefore unable to help our economy grow. In a sense, when we help them, we are helping ourselves.
Thus, I would say that it is fair to say that we may believe somewhat less in individual responsibility. However, to say that this is no longer a “key” part of our political culture is simply not true. It is an argument used for political gain by certain parts of our population. I believe that the vast majority of Americans still believe very strongly in personal responsibility.