This part isn't exactly true, at least not in NC. This is not to say that it wouldn't be exceedingly difficult, but I don't think it would be impossible.
Here's why. When someone requests an absentee ballot here, a unique code is generated for that request. This code can be found printed on a label which is affixed to the return envelope for the ballot. Said label also has the name and address of the voter who requested the ballot.
There is a box on the ballot itself where that same code is copied (by hand, which may offer even different issues).
So there are two separate items which if combined can provide a direct trail to the identity of that particular ballot's recipient.
Now I suspect (would hope) that there are layers of security which are intended to prevent, or at least make it exceedingly difficult to manage to combine these two data points, but nonetheless they are there.
I have no idea about the process in any other state, but that is what has been done here for at least the last three elections. (Those being the ones for which I have requested an absentee ballot.)
And I can provide photographic proof, since I chose to vote during the early voting period, and still have my original absentee ballot package in its entirety.