They run off the alternator. The battery is there to act somewhat as a “filter” but most vehicles will happily sit there and run without a battery.Not true. The vehicle runs off of the battery not the alternator. The alternator just keeps the battery full.
Even if it was true, if you increase the load on the battery the alternator would have to work that much harder to recharge/maintain the battery putting us right back to “increased load=less mpg”