Visible crew/equipment: During the jungle chase, Marion jumps out of the truck, leaving Dovchenko driverless. A few shots later, Spalko fires on the two amphibious vehicles, and when Mac removes his cap to check his head, the actor's modern wired device (earbud or mic perhaps) drops out from under his scarf/shirt and hangs over his shirt. Later, in the chamber with the crystal skeletons, Mac admits to lying about the CIA, then as Spalko climbs the chamber stairs, it looks like her lav mic is a bit visible between two upper buttons. (01:17:10 - 01:44:15)
Answer: People felt it was ridiculous and cartoonish, even by Indiana Jones standards. Even if it was possible to survive a nuclear blast via the lead lining of a fridge (it's not), or that the fridge would simply be thrown away rather than be melted/torn apart like everything else in the vicinity (it wouldn't), the impact of being flung what appears to be a mile or so through the air, then violently crashing into and rolling over the ground, would certainly kill anything inside. The controversy arose because usually, in "classic" Indy films, the fantastical elements were exactly that: fantastic, magical, and/or supernatural. This was presented as taking place in our reality, with no "power of God" or magic spells, and for many, that was just too much disbelief to suspend.