Hmm. I think they're deluded but not dumb enough to cut his throat in the truck and cause the resulting mess, which would require more cleanup than just yanking the sweats. They wanted that truck, they weren't going to ruin the interior like that.
For that matter, I also don't believe in the Fargo/wood-chipper theory. (Not to say I don't believe the wood chipper was stolen).
I do not have an alternate theory of why LE knows with precision when he died, but the classic cars we've seen in the hangar were immaculate. Obviously they valued these vehicles over human lives, I don't think they'd do something that would require the entire interior to be replaced down to the floorboards.