I'd venture to say that most YC rejected companies (that would define themselves as successful) could care less about proving anything to YC. Building a company is hard and stressful and beating yourself up over something that happened in the early stage is foolish. It usually comes down to not pitching it well enough or being unlucky. A "successful" company will overcome much bigger obstacles to get to that point.

It's a two way street and sometimes the company may be happy they didn't get into YC because it opened different doors that led to success. Maybe in some cases YC could have closed some doors that were crucial.