My problem is that I do not see the difference between poofology which produces incomplete duplication, or ToE which does it equally indistinguishably.
Traditionally we call this philosophical view "Last Thursdayism", since the
reducto ad absurdum of your view is that God not only specially created all living organisms in such a way that they bear a (false) appearance of common descent, but that he also created fossils in the ground, the light of distant stars in mid-route to Earth, and indeed your very body and (false) memories, all at approximately 3 o'clock last Thursday.
Truly, there's probably no experiment we can make that can distinguish between a universe actually as old as it appears to be and a universe created last Thursday with an apparent age of 14 billion years. Traditionally the Last Thursdayism view is rejected on grounds that are either largely aesthetic (people just don't like the idea of a God engaged in a colossal deception) or on the basis of an
a priori rejection of the existence of God.
The seam in the argument, in other words, is that it's too strong - it proves way more about God than the Discovery Institute is prepared to accept.