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I have to point out the VLS launcher has a seriously weird all-solid-fuel design. I am not a rocket scientist, but, as an engineer, I have to wonder what the hell those rocket scientists knew no other spacefaring country did since nobody launches satellites on all-solid-fuel rockets.Sure they do.
For example, NASA's LADEE mission was launched to the moon in September 2013 on a Minotaur V: a five-stage all-solid rocket.
Last month, India just launched a Mars orbiter using the PSLV: a 4-stage rocket with only the second and fourth stages being liquid. The first stage was solid, and had 6 solid rocket boosters. Thus, it launched entirely on solids.
And of course, there are lots of liquid-fueled rockets with solid boosters on liquid-fueled rockets. The French Ariane 5, for example, has one liquid core stage plus two solids. The solids produce 92% of thrust at liftoff. Solids are great at producing thrust.
The American Titan IIIc rocket took this configuration to its ultimate conclusion. It was launched solely on the solid rocket boosters. The liquid-fueled core stage did not even ignite until a couple of minutes into the mission, i.e., although it was at the bottom of the stack, it actually served as a second stage.
If you're talking about spacefaring nations that avoid solids, you're really only talking about the Russians, Chinese, and Ukrainians. They only use solids on military missions, and stick with liquids on civilian missions. They get the necessary thrust by clustering lots of engines together.
The Chinese moon rover, for example, was launched on a Long March 3B with a total of 8 engines ignited at lift-off -- 4 liquid-fueled engines in the core stage, plus 4 liquid rocket boosters. Yes, that's right, even their strap-on boosters are liquid-fueled.