It’s possible Indy knew the Biblical reason for looking, which the audience would have to know as well:
1 Samuel 6:19 King James Version (KJV)
19 And he smote the men of Bethshemesh, because they had looked into the ark of the Lord, even he smote of the people fifty thousand and threescore and ten men: and the people lamented, because the Lord had smitten many of the people with a great slaughter.
Indy says something snarky to the government agents at the beginning like “didn’t you guys go to Sunday school” or something like that, so I think it’s safe to say he would have been familiar with the ark lore, including the stuff in the Bible.
I think you’re really correct here, though-that deleted scene was really how they really intended to get that info across, but I suspect that cutting in order to prevent too much foreshadowing.