Paul does not want his own righteousness that comes from the law, but that which comes through faith in Christ. This is the core of the New Covenant: righteousness is a gift, not an achievement.
There is a deep difference between a righteousness you work for and one you receive. One makes you tired, the other sets you free. Paul, the former Pharisee, knew both. And he chose the gifted righteousness.
Today you can stop earning your standing and start receiving it.