One of the reasons they're not as common anymore is because of the dangers associated with them, not only in earthquake country, but also fires.
5-over-1 refers specifically to the wood construction of the top 5 floors. You can have a soft story building that is not built out of wood and even more vulnerable to earthquakes.
> The name derives from the maximum permissible five floors of combustible construction (Type III or Type V) over a fire-resistive Type I podium of one floor for "5-over-1" or two floors for "5-over-2", as defined in the United States-based International Building Code (IBC) Section 510.2.[1][4] Some sources instead attribute the name to the wood framing of the upper construction; the International Building Code uses "Type V" to refer to non-fireproof structures, including those framed with dimensional lumber
From the article: "5-over-1 construction—so called because it frequently combines up to five floors of inexpensive wood-framed construction over a concrete podium—is popular primarily because it is a cost-effective means to build, especially here in timber-rich Oregon."
It's a mostly new style.