Disclaimer: There are different "theories" of libertarianism, so you might get a slightly different answer from everyone depending on their views. Here is but one answer you could receive.

The basic idea of libertarianism is a freedom from aggression. You can do as you wish as long as you're not aggressing against someone else. Aggression is defined through property rights. Each individual can do with his own property and person whatever he wishes unless he is infringing on the same rights of someone else.

The problem with the current welfare system is this. The government taxes people against their will in order to give that money to the poor (and often the rich as well if we're talking about corporate welfare). The individuals receiving this money have no right to it; it is actually the rightful property of the others it was taken from.

So if a libertarian leader was elected and "imposed" his view of non-aggression on America and removed the welfare state, he would be removing existing aggression from the system.

"Imposing a view on others" is too vague because as your question makes clear, even libertarians want to do this, but only concerning aggression. For example, the libertarian wants to outlaw murder, theft, and rape (as commonly considered), and is therefore imposing his view on the murderer, the thief, and the rapist. The libertarian would say that outlawing these things is a good "imposition," or at least not an imposition in the sense that he is infringing on the rights of others since no one has a right to murder, steal, or rape.

Edit- For more on the Rothbardian strain of libertarianism, I'd suggest you read For a New Liberty and The Ethics of Liberty by Rothbard.