The bird was found on an airlinerIts cage got broken as they cleared the plane and it flew through the gate. They caught it without any bad incident and everyone fondly remembers that day.
They brief the passengers they takeThe passengers were told why they had been taken and they all understood it. That's why they calmly shuffle to the gate when told to at the end.
Using Time Travel to stop whatever polluted the world has already been tried, and it failed.They decided to risk the timequake and remove whatever fouled the Earth, but it didn't work. They can't go back to those times now to try again.
J. Edgar Hoover re-established the Millennium Group.The retcon of Hoover founding the group instead of it existing since the Middle Ages can be explained. The group was likely shut down by the House of Un-American Committee or Joseph McCarthy, accusing them of being communists. Hoover, a member of the group beforehand, re-formed it as a "criminal investigative consulting firm" after the Communist Witch Hunt died down.
The Frenchman was a member of the Millennium Group.He has the same visions as Frank. He left the group, believing he was given a mission to kill people he thinks are evil.