Let's take a look at the question: "Why is reaching the lost important?" First, who is "the lost." That's pretty judgmental to assume that some one or a group of people are "lost." What does it mean to be lost? In a literal sense, it means some one who doesn't know where they are, or how to get where they want to go, or maybe they don't know where they're going. In Christian circles, I've heard some say it means "people who don't know Jesus." I thought this for quite a while, and still do to a degree. The more I think about it "lost" isn't a judgment of someone's character or pre-eminent eternal destination if they're not "found." I tend to think of lost as "naive" or part of unjust systems without knowing it. In this sense, we are all "lost"...almost hopelessly. We all participate in unjust/sinful systems without knowing it, and many people bury their heads in the sand. We go through life trying just to be "ok" and survive with some enjoyment here and there. Occassionally, we may become aware of how we participate in unjust systems, but we find a way to push it aside, deny it, and get back to being "ok." Could this be what it means to be lost?
I think most people come to a place where they realize just how lost they are. You get to a point in life and you think, "How did I ever get here?" Then we find a way to get back to "ok." We buy some more stuff, or change jobs, or move to a new place. We change our relationships even through difficult things like divorce. All trying to find some kind of existence that's "ok."
What if, Heaven is real, and there's more than just "ok"? What if all of this stuff we're using to feel "ok" is keeping us distracted from the reality of the kingdom of heaven? What if we've been led astray by shiny things that promise to make us "ok" when really, there's abundant life breaking in to every moment all around us? This abundant life isn't things that decay and fade, but things that last: love, joy, peace, hope.
For me, reaching the lost is urgent and important, not because I'm afraid God is angry and going to punish and torture them eternally, but because with each breath, and each day that passes is a day they're missing out on The Kingdom of God. It's another day they're missing out on The Abundant Life (John 10:10) that Jesus brings us. It's another day of missing out on the deep joy of life with Christ. It's another day of missing out on the celebration of heaven.
I don't want anyone to miss out on another day of life with Jesus Christ. I don't want anyone to miss out and live another day without the radical, extravagant Love of God in Jesus Christ. I don't want anyone to miss out on another day of 'real' heaven. Heaven on earth that is already here, but not yet complete. That's why sharing my faith and showing God's love is important and urgent.