In our culture in the US responsibility is typically associated with adulthood. The more of it you have the more "grown up" you are. And you've probably heard phrases like: A boy becomes a man when a man is needed.
And there is a good reason for that. I've come to find that responsibility is essential for self-actualization as well as maturity. There is no success in relationships, in business, or in life without responsibility. By definition if you have no responsibilities, if nobody relies on you, then you don't matter. If you disappeared nothing would change and nobody would notice.
Also responsibility is found to be very psychologically rewarding. This part of the reason why people love having children and doing volunteer work. Being selfless is intrinsically fulfilling.
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Now I share in the sentiment that there isn't much value in being responsible for stupid crap like paying bills or creating TPS reports. But being able to choose your responsibilities (a noble goal) is not the same as avoiding responsibilities (immature and selfish).