I should have been more specific. Babylon was dominant for 70 years, in the way that the ancients reckoned. Some of the upper echelon were taken to Babylon at the beginning and again later, and some returned when Cyrus told them - including from other nations - to return. Many, as you point out, made a good business (Egibi Bros.) and stayed there.
The Biblical text of Cyrus' letter (not really a "decree") is not a replica of the text on the Cyrus Cylinder. It's a religious rewriting to suit the agenda of the later writers (Chronicles and Ezra).
The names of the people recorded in their Scriptures as leading the initial returnees to Yehud were likely concocted to make a link to Jehoiachin.
This marked the time when the priestly class's desire for power was able to be mounted, now that the royal household was gone. A new entity emerged and they later became known as "Jews".
There was another major change following the destruction of the second temple, although unlike the Jewry witnessed today.