How many of those space agencies can really operate as fully independent space agencies?
Considering that Russia beat the US to almost every single milestone up until the Moon landings, and then we were in a cold war until the 90s, that's a pretty good case for them being completely independent of NASA.
Helping other countries peacefully explore the solar system seems like a more reasonable explanation for NASA being involved than trying to somehow control or manipulate a dozen other agencies of various countries with false data and lies.
China certainly isn't a NASA puppet, and they are highly capable and would have certainly noticed if NASA has been lying about what happens to rockets when they go up. They would have used that against the USA, think of the propaganda opportunities if they exposed NASA as a fraud.
I don't see any reasonable evidence that NASA is controlling the rest of the world's space agencies and governments.