Mr. Ping is the reason Po is the dragon warrior.

This is a very perceptive observation! Yes, I think that there were at least two major factors that allowed Po to become the Dragon warrior. One of those factors was Po's own inherent character. The other was Mr. Ping's influence.

It is interesting that Shifu (unintentionally) managed to make some very major mistakes when it came to raising his own children, Tai Lung and Tigress. In Tai Lung's case, Shifu loved unrestrainedly, but with too much pride, and with expectations that ultimately could not be met. With Tigress, Shifu closed himself off emotionally, but there was still that expectation (I think from both Shifu and Tigress) again, that she would be the Dragon Warrior. Shifu had good intentions both times, but it just didn't turn out well.

Mr. Ping is a very different sort of person. He loves Po completely, as a good father ought to do. Po felt so completely and totally accepted by Mr. Ping that it never seemed to occur to him at all (until the second movie) that Mr. Ping might not be his real dad. Mr. Ping has always had the understanding that "to make something special; you only have to believe that it's special." Po was always special to him BECAUSE Mr. Ping loved him. Po never needed to be anything other than himself.

I consider this significant, because I am convinced that Tai lung and Tigress never felt accepted in the same way when it came to Shifu. Once again, I am sure it was never intentional, Since Shifu is a good person (albeit flawed) who wants to do the right thing. But the expectation of both his children/students to prove themselves by becoming the Dragon Warrior, may have given them the impression that the only way they could truly be loved an accepted by Shifu was to be the dragon Warrior. It was their goal, it was their purpose, and they may have felt that it was the main reason they had been adopted in the first place. This would have made it impossible for either of them to be the Dragon Warrior. Can you imagine what poor Tigress would have thought if she had been the one to open the scroll? Or Tai Lung?

Mr. Ping really did a wonderful job raising Po. That does not mean that Po was free of his own issues regarding his self esteem. He was loved and felt loved, so that part of his life was not lacking. It was his wish to somehow be better than what he was, his desire to be 'more', may actually has been due to his fate as the Dragon Warrior. His wish to rise above his lot in life led to the very actions that resulted in his being chosen as the Dragon Warrior in the first place.