You nailed it.
That is exactly why it is a criminal offense to remove classified documents from appropriate designated areas.
This is day one training for any civilian or military hire.
It is not just classified documents—you are not even supposed to remove any documents with personally identifiable information (like social security numbers, for example) of anybody from secure areas.
And what is even MORE outrageous is that he took these papers when he was a Senator and a VEEP, which is doubly ILLEGAL!