You're right: It doesn't make sense for people to turn their back on God when they obviously need Him -- yet it happens all the time. God loves them and wants to help them, but they still stubbornly refuse to turn to Him.
God may even be using these circumstances to try to get their attention -- and yet they still refuse to listen or turn from their ways. They are like the people of Jeremiah's day: "They did not listen or pay attention; instead, they followed the stubbornness of their evil hearts" (Jeremiah 11:8). Sadly, because of this, their nation was eventually destroyed.
What is the problem? Why are we like this? The problem can be put in one word: pride. We think we can get by without God; we may even think it's a sign of weakness to admit we need Him. But pride blinds us to our own faults, and it also makes us stubbornly cling to our own ways. But pride is a sin; in fact, it's at the root of almost every other sin. The Bible warns, "Pride goes before destruction, a haughty spirit before a fall" (Proverbs 16:18).
Pray for your cousin; only God can open his eyes and convict him of his need of Christ. Also, make sure you put Christ and His will first in your life every day.