There is nothing mysterious; the sauce does not hold the food together or keep it warm or some other such magic. Instead, the reaction you are getting is some combination of:
- The people serving know how they believe shawarma should taste, and whatever this sauce is is an integral part of that (likely in texture as well as flavour; the shawarma I've had would be very dry without sauce).
- Especially for food associated with a minority culture (this is shawarma being served in Canada), people can be resistant to changes.
- The specific flavour is also tied to the identity of the food: shawarma, gyros, souvlaki and a doner kebab are all superficially very similar food items (roasted meat in a flatbread with some sauce and salad) but the specific details are what make them what they are.
- They probably constantly serve people who ask for changes and then complain that the food is bland or tastes wrong, which means they want to make sure that you really mean it.
- As with many non-corporate-chain fast food places, they are probably used to a fairly relaxed, jokey style of banter with customers. "Are you out of your mind!" might sound more extreme if you're not expecting that, but is likely just teasing.
Analogous requests (from the respective servers' perspectives) might be asking for no 'red sauce' on a pizza, no 'yellow sauce' on a hot dog, or no 'green sauce' with nachos.