In various forums with Tesla owners in similar climate conditions, people agree with Broder's observations. He didn't "lie" about what happened. It's something they noticed as well.
Now you can carp on petty points like him using common sense, but when a car company lets you test drive a car and they give advice, you should follow said advice. I understand crying foul if he did something ill-advised, but Musk did not refute the fact that they gave him what seems to be poor advice. And I don't blame Broder for following what Tesla said, even if it ended up with a picture of a Tesla being towed away.
I'm very surprised Tesla hasn't released the call logs, to be honest. And I suspect, if you saw them, you would agree that Broder's actions are in line with what Tesla told him while he was in the car
"And your log data makes me want to buy one of your cars. Thanks for that marvel of engeenering."
P.S: I recommend you read some of the comments here. The data is in-line with Broder's account.
This is active disagreement over whether Broder was cleared to make that last trip. Musk says it was obviously no, Broder says yes.
That is a very simple way of resolving this. And completely within Tesla's and Musk's ability (and if Broder really was authorized to do so, he'd let them release the call logs)