It teaches them independence and agency.
As far as I can tell, my choices for dealing with injustices and terrible coincidences in life are:
1) to assume someone else is responsible to solve the problems for me, or
2) to assume I am responsible to solve my problems.
The former winds up leaving me bitter and angry with others in my life for not doing their job and taking care of my problems.
The latter helps me develop a sense of agency and helps me cope better with injustices and uncontrollable hardships in my life.
In this perspective, "fault" means "who caused this problem?", while "responsibility" means "who should try to solve the problem?"
Thus, taking responsibility for your problems doesn't mean assuming guilt for their existence.
This also applies in an inverse way to fortunate coincidences - in your lottery example, this mindset would say "I did not earn this money, so I'm not going to take pride in having won it, but I am responsible to use this money well."
This attitude also doesn't mean absolving wrongdoers of the harm they've done. Taking responsibility for healing the hurts others have done you may often include telling third parties about the harm and seeking help in repairing the injustice and preventing similar ones for potential future victims.
As that implies, "taking responsibility" for your problems also doesn't mean solving your problems solo. If help is offered, take it! If you need help, ask for it!
Just don't sit around waiting for someone else to deal with the hardships and problems for you, because that leads to despondency, disempowerment, and hopelessness.
That's how this has worked in my life, anyway. I really wish I'd been raised with this idea - I didn't encounter it until my mid-thirties, and it was a significant discovery for me.