The records of those visits, as you know, were destroyed during the so-called break-in so I don’t know how it would ever be proved. The old argument that he couldn’t travel to Pakistan on a U.S. passport was debunked - there was just an advisory against U.S. citizens traveling there at the time, not a ban on travel. So, while you are correct, the evidence is probably gone.
The evidence cannot be all gone. Therea are recoverable hard drives, for one thing, there are people who know and may have copies of their own, there are records of entry and exit from not only the US but the other countries.
There is too much paper for it all to have disappeared. Way too much.
And the very fact that he admitted he travelled to Paki and Indo and IIRC also Hyderabad, India (there is a Hyderabad, Paki so it’s a little unclear perhaps about that part), and definitely did not have a US passport at that time, at least in the name Barack Obama, then he obviously used a passport of another country.
He also used another passport when he traveled as a child from HI to Indo and back again. But the lack of US passport when traveling as an adult is 100% damning even without the records in hand.