Yup, Amanda Waller decided the world needed another Batman, and took it upon herself to recreate the conditions that led to the first one

She has a long history of putting villains to work on questionable schemes for "The Greater Good"

Of course the lab rat bit was incredibly convoluted. They modified Terry's father so he would pass on Bruce's DNA rather than his own without being aware of any of it

Then Waller tried to get the parents killed in front of Terry, so he would have the same trauma that led to Batman in the first place, but Phantasm backed out and refused

Yea but why Phantasm out of all the villains? And in the clip I posted was the Barbara Gordon the older one? Then she had to know what she was getting herself into. So basically your saying (and not looking for an argument) in some cases One Bad Day is actually true what the Joker said?

Not sure I know who you are asking about. The woman speaking is Amanda Waller (the same character in charge of The Suicide Squad). Barbara Gordon did not appear in the clip, but in this continuity she grew up to be police commissioner.

As for why Phantasm, I assumed it was because the story called for someone who might agree to do it at first, but would ultimately refuse to kill them on moral grounds. It didn't strike me as the most calculated choice in-universe, but it was a suitable character for the story they were telling.