I'm not even sure which post to quote to better illustrate my point, but; about how Vader and Palpatine never went to seek out Luke and Leia: Palpatine told Vader at the end of ROTS that he had killed Padme, right? Right.
Padme hadn't yet given birth when she was taken from Mustafar by Obi-Wan (which was before Palpatine arrived). So it's likely the dynamic duo simply didn't expect Padme to be alive to actually give birth; if they would have then surely Palpatine would have gone looking. Vader was quite the good servant, but a new-born Skywalker would no doubt hold much greater promise.
So, the children were free to grow up; hiding in the shadow cast by one of Palpatine's very rare oversights.