The question I have then... why would they choose a real person to make up a story. A wonderful story no doubt about that, but still a made-up story.

There is not much story in New Testament. We don't even know what Jesus do as a God in human flesh for decades, except that he could be a carpenter as His foster father. So, for a made-up story it is too specific and Gospels looks exactly like journalist reports by different journalists about event, not a story about person Himself.

So, they killed him and then decided to write a story and a narrative they (the Empire) could control.

The Empire didn't control the "story" at all. At first, Romans account followers of the "story" as a strange bu harmless small community, then as a threat to Empire and tried to exterminate all of them, and finally Rome itself fall into the "story". That's not how controlled narrative work.

Apart from theology, New Testament is basically is an opposite to the Old Testament story that glorified a single tribe and justified their crimes against humanity. Both parts become a perfect example and severe warning for all humanity of why and how to not become Jews.

It is not hard to imagine that Jews will always try to undermine such manual, creatung "new translations" and "new interpretations". And if you add a theology, then it become obvious why Jews always tried to create a narrative that Christianity is something weak, boring and controlled.