I would think that the appliance amp draw upon start up would be a problem for consumer class generators.

For example, when a fridge or freezer compressor kicks in, the amp draw is many times higher then once running. For this to auto start the generator, the battery at the generator would need a 12 volt inverter to briefly handle the appliance start load until the generator started and came up to speed.
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Earth and sky, woods and fields, lakes and rivers, the mountain and the sea, are excellent schoolmasters, and teach some of us more than we can ever learn from books.

John Lubbock