The compressor runs off the engine because a 12-volt DC motor that's powerful enough to turn it (if it's possible to make one small enough to fit under the hood) would be more than the alternator could handle and would drain the battery in minutes. Why not put in a bigger alternator to handle it? Well, then you're back to a power-robbing device running off the engine again. I believe it's probably more efficient to power the compressor directly; at least it's able to cycle on and off as required instead of being engaged all the time.

The blower motor runs off the current supplied by the alternator (or the battery when the engine is off). It doesn't make much - if any - difference on fuel usage.

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My religion consists of a humble admiration of the illimitable superior spirit who reveals himself in the slight details we are able to perceive with our frail and feeble mind
Albert Einstein