Partially true Kevin.
Actually, a unibody structure can be as heavy, or heavier than a body on frame vehicle. That was the case with Chryslers early unibody cars. Even the compact, Valiant and Lancers. They weighed considerably more than GM's body on frame designs from the era.
The real advantage was cost saving for front wheel drive. Also robotic welding.
GM slashed 800 pounds from full size cars in 1977, and 600 from intermediates in '78, while keeping the separate frame. It costs more to fabricate a frame. It entails not only welding, but folding, and in later decades, pressure forming with water or hydraulic fluid. So, with the cost to re tool for front wheel drive, it was less expensive to have robots weld the unitized space frame, and forget the extra full front to rear bumper rails, and cross members. Sub rails for unibody cars are usually only 3 dimensional, not 4, at least since the GM F body died. Front drive power train cradles are 4 dimensional, but smaller.
Range Rover's have been the exception, until last year. The Discovery and Sport versions were both unibody, and on a full perimeter frame. The Discovery or Sport may still be.
Not trying to sound like a know it all, but I knew a few engineers because of where I live and worked.