It depends. The battery issue is a nonissue, but the short trip can absolutely have an effect. It takes longer for your engine to reach true operating temperature than the oil/water coolant to get to temp. There’s videos out there that can explain the equations for larger engines, but if the drive is less than 15 minutes, it’s likely he’s building up carboxylic acids from nonvaporized water in the pistons. That ain’t a good thing.