if one develops such an asking mentality, he or she won't go far in life because all his thoughts will become shallow;

Spot on! I have seen people with such mindset keep on being stuck at the very place throughout their lives.

On the other hand, there are some whose ego never let them ask for help even when they are in a dire need.

All of us need help at one or the other point of our lives. Important is that we are careful about whom we are asking to. I believe, there are some genuine and trustworthy people in our lives, who never want us putting down