As for the case of CP3, not the first time, we have seen a lot of great players not winning a ring, like Charles Barkley and yet they are still in the discussion as great players. Perhaps the closest that CP3 could have won is with the Rockets, but he got injured and so they lost that series against the Warriors and then the Dubs winning that champion that year. They still have a chance next year, but it was a disaster for them.
The fact that Paul signed a one-season contract with the Spurs this offseason indicates that he has probably decided to give up trying to get at least one ring. Obviously, he won't be able to do that with Spurs in the upcoming season. Besides, he will be 40 years old next year, so I wouldn't rule out that the upcoming season could be the last of his career.
Indubitably, he already made a shot with Warriors and Suns but failed to achieve it because he always choke on the crucial matches. He has a lot of chance to get a ring in his prime but I doubt he can get it anymore without joining on super team that can win a champ without his major contributions such as Nuggets or Boston.
Although he can provide a good contribution to Spurs since Wemby can finally have a teammates that will keep passing him the ball. Spurs young players play on their own just to get the spotlight. Paul addition to their roster will give more support focus to Wemby which Pop will probably capitalize.