Looks nobody noticed yet, or didn't pay attention.
Do you remember the file with Bogdan Khmelnitsky letters Putin gave Tucker?
Now think - why is it so thick and heavy? Copies of all Khmelnitsky correspondence could not be so thick and heavy.
What else was in that file?
Yes, Tucker had no other option to hand over important documents than during the recording of on an interview.
Makes perfect sense.
Handing over something with stuff around is insecure. Disappearing from their view in such a place like Kremlin is also questionable. You can't be shure for 100% in other people.
Also, you might have missed a smile in a title.