You likely talked to the service manager at the dealership. Very few of these people have much in the line of mechanical abilities and rely on their mouth to get through the day.
He said he didn’t believe they left the cap off. How does he know? He wasn’t the one working on the vehicle.
Yes, it’s entirely possible to go that far with the cap loose. At times the cooling system may lose a spoonful of coolant here, a half a cup there, and eventually the coolant level is low enough to cause overheating. When this occurs the loose cap may be blown off.
Someone screwed up by failing to tighten the radiator cap and they are going into their CYA (cover your axx) mode.
This guy may claim to have started out in the service dept. but that doesn’t mean that he ever had a clue either.
What would I do? I’d go in and ask to speak to the dealership’s owner about this problem. Keep it firmly polite at this point as the owner probably has no inkling about what happened here. Tell the owner you hold them responsible and intend to bring a suit in small claims or district court (depending on the law in your state).
Just for grins, before approaching the owner you might tell the service manager to his face that in 2 minutes you will be speaking to the owner about this matter. Watch his reaction.