The Short Version
The claim is largely accurate. It is well-established in automotive engineering that cold starts cause elevated engine wear due to insufficient oil circulation, loose metal tolerances, and fuel dilution — and short trips multiply cold-start frequency per mile driven. However, the claim oversimplifies: the severity varies significantly by oil type, ambient temperature, engine age, and vehicle design. Modern synthetic oils and engine management systems have substantially reduced (though not eliminated) this effect. Idling to "warm up" is itself counterproductive; gentle driving is the recommended approach.