I was always told that on carb'd engines, that carb heat should always be first, as the heat will drain quickly from the engine, and if you leave it until the end, it may already be useless.
What ever about the other checks, I know if I have a engine failure, my hand will go automatically to the carb heat. If the engine makes even a slightly different tone, my hand goes to the carb heat
And now that I spend most of my time flying injected engines, I'm even more nervous of the carb heat, when flying one with it.
dp