Counter arguments, a hotdog is a sausage, the bread is a condament. When you buy hotdogs at the store you’re taking about the pack, when you’re cooking a hotdog, you’re taking about the sausage being cooked. A hotdog on the grill is not in the bun. When you’re eating a hotdog without a bun, you’re still eating a hotdog.
In the other direction, a hotdog with mustard is still called a hotdog meaning the mustard has no say in the state of the hotdog.
Furthermore, we have splitlink sandwiches so a sausage as sandwich still needs the sandwich modifier. When I say “hotdog sandwich” it’s bothersome because it conjures the idea of hot dogs between two slices of bread.
So if a hotdog is a hotdog with or without the bread, and a hotdog is a hotdog with or without the mustard, than the bread plays the same role and becomes a condament for the eating of a hotdog that belongs firmly in the category of sausage.
Spare points to back this up is taco, chicken taco, fish taco, street taco, all need the modifier “taco”. If I say we’re having fish and serve a tuna taco, I’ve not given you the accurate information. The same goes for wraps, without the “wrap” modifier you get different information. In reverse, we do not ask for a bun to get a hotdog. Following along that line, we have split bun sandwiches which use a bun and are not explicitly hotdogs.
Lastly, with this information you can order the incredibly cursed, split link split bun sandwich with mustard which presents as a cut hotdog with mustard but is in fact an entirely different thing all together.