Jesus is not cold-hearted. He does not want to condemn anyone. He does, however, recognize that the world is a very big place, and God's work is a huge undertaking. The reality is (for Jesus and for us) that we cannot possibly save everyone. Look at what Jesus is asking his disciples to do. Go everywhere they can, proclaim the good news of a loving God with the challenge of change, welcome the ones who welcome them, and move on when people don't. There's just too much to do to spend valuable time and energy with people who do not want to be saved.
This is one of the most difficult things I've ever had to learn. First, that I cannot save everyone - in fact, I cannot "save" anyone. Saving is not my job. Proclaiming the one who saves is. Second, that I cannot even make anyone listen to the proclamation. They either listen or they don't. They change or they don't. They choose life or they don't. As desperately as I want to save people I love, I simply cannot. But I can pray. I can be there for them in whatever capacity they need. I can turn my attention toward people who do want my help. I can accept help from people who offer it to me.
Then Jesus called the twelve together and gave them power and authority over all demons and to cure diseases, and he sent them out to proclaim the kingdom of God and to heal. He said to them, “Take nothing for your journey, no staff, nor bag, nor bread, nor money—not even an extra tunic. Whatever house you enter, stay there, and leave from there. Wherever they do not welcome you, as you are leaving that town shake the dust off your feet as a testimony against them.” They departed and went through the villages, bringing the good news and curing diseases everywhere.