Capitalism makes the same mistake that socialism does. It attmepts to negate the human part of the equation. In order for true socialism to work it must negate individuality, creativity, the quest for excellence and personal satisfaction from its equation. On the other hand, true capitalism doesn't take into account greed, man's inherent need to dominate, Darwinism and the lack of society.
Each tenet is destructive to a society. They just attack from opposite ends. Neither a truly capitalist society nor a socialist society can survive the needs of humanity. Capitalism has just as many non-human traits as does socialism.
Anyone who thinks that capitalism can solve all problems thinks that he is the biggest bear in the river. He will get all the salmon and survive and to hell with everyone else.
Capitalism is not crippled by greed because capitalism is regulated by the free market. When people are free to buy and sell as they wish on any mutually agreed terms greed gets its come-uppance. There are numerous examples of greedy companies and individuals who are no longer around because the free market (not the government) weeded them out. You say capitalism ignores greed and darwinism but greed is just another factor that darwinism punishes in a free consumer society.
Alternatives exist. One alternative is Distributism, the economic state where the possession of land and capital, instead of being concentrated in the hands of a few, are maximally distributed among all men throughout society -- a system of communitarian free-enterprise economics based upon the medieval Christian social order that relies on widely-distributed capital and guild regulation of commerce as the basis of economy. Distributism was championed by G.K. Chesterton, Hilaire Belloc, and many others, and its principles were affirmed by His Holiness Pope Leo XIII in his famous 1891 encyclical Rerum novarum.
Unfettered capitalism leads to the same place unfettered communism leads: a condition of widespread misery, tyranny, and injury to the dignity of human life. Those seeking an economic alternative with a human and Christian character should explore Distributism.