By and large Jesus focused on living the way he wanted. His teaching about hell was used as a way to wake up believers who weren't showing fruit. It wasn't pointed at non-believers. Similarly, Paul said it wasn't our job to judge non-believers, and in the first few chapters of Romans could even be understood as saying that outsiders might be members of God's people by following their conscience. With the sheep and the goats, the people who are saved are surprised. They didn't know they were serving Christ.
Some Protestants immediately say "Salvation by works!" But that's not the point. Jesus taught that we should base our actions on love for God and others, not on trying to earn credit with God. Paul's attacks on works really was directed against legalism, "works of the Law." Both Jesus and Paul said that we are accepted because God loves us and Jesus died for us, not because we're earned his favor. I think it's perfectly possible for a non-Christian to do the right thing because it's the right thing, and because he cares about other people, and thus avoid the kinds of problems that Paul meant by works righteousness. You are absolutely right that many unbelievers have the same kinds of experiences that we have.
So does Jesus' death matter? I think so. First, like leaven, Jesus' followers really have changed the way people throughout the world deal with each other. Modern secular humanism is really based on Jesus' teachings, and even on his self-sacrifice. The idea that true glory comes from giving ourselves, not from possessions or power, has had a very wide effect.
Second, I believe in spiritual connections, of people with each other, and of people with Christ. My understanding of Paul's doctrine of the atonement is that Jesus death changes us because we die and rise with him, and our union with Christ transforms us. I don't think either spiritual union or the power of Christ's resurrection are limited to Christians. I do think that Jesus is the source, and we're a lot better off as near to the source as possible. Many of the places where evangelism is most needed and most difficult are exactly the places where the spiritual influence of Christ has been the least active. Our role as "leaven" (as Jesus calls us) is really important.
I am not a universalist. I believe people can get into spiritual states that make it impossible for them to participate in the eventual kingdom. It would be really convenient if no Christian was ever in such a state, and all non-Christians were. Things are not so neat. Yet I do believe that the power of Christ is important, our our role as the Church in being the leaven for the world is important.
