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.

I've owned many cars and ran a van hire company.

Look at the owners manual in your car, and it will say trips less than a couple of miles are considered severe duty, and the oil needs to be changed more often. Your can get condensation in the engine that can break down the oil faster.

Cold starts are where most of the wear and tear occurs.

I believe it comes from a saying that most wear is in the first ten minutes of driving.

Which I also believe assumes you don't "wait for the car to warm up"

That saying. I still find some truth to it. While modern cars can adjust fuel mixture to different conditions. There still is an unavoidable few minutes where the engine runs in an open loop and wear is greater.

That said, in my vehicle when I listen for the engine to "close" it happens in less than 5 minutes.