They don't have to Soulcast the wood into stone (though I imagine they could, with some thinking). They could slowly Soulcast from thin air.
And it's mentioned that each highprince was responsible for the bridges closest to them. I don't know if that was official, or if they just protected their route to the plateaus. But even if there was no unified effort led by the King to establish Soulcast bridges, I would still expect to see a highprince try it out (especially Dalinar). That and the Soulcast outposts.
Those structures are built on a foundation, though. The bridges stretch over empty air.
For a good comparison, look at how they used to construct buildings out of solid concrete. They could withstand an impact from an airplane (look it up), but they tended to be a safety nightmare in case of earthquakes.
A good point, but look at the Roman Aqueduct then. Still standing,
still functioning, after ~2000 years. Built on a foundation, but much more like a bridge than a building. I'm sure it's seen a few earthquakes in its time.
And surviving impacts like boulders or Parshendi attacks is a more useful feature than surviving earthquakes. They can always repair the bridges after the rare earthquake.
I'm also starting to think that Soulcasting a bridge might not be possible (under current conditions). Given the circumstances (Parsheni attacks/raids, highstorms), the time it would probably take to Cast, and that Dalinar would probably have tried to implement it already (he only one not thinking about wealth, and he's always concerned for the well-being of the bridgemen), leads me now to think that as long as there's a war on the Shattered Plains we won't see Soulcasted bridges.
They don't have to Soulcast the final assault, or even any distant plateaus. They can go out at night (since most Soulcasting is done at night to hide it from the common soldier), and Soulcast the plateaus close to camp. Then move outward. When they're far enough away that they start to worry about Parshendi raids, they can send a force with mobile bridges to hold the surrounding plateaus, and Soulcast in safety (and if privacy is still a concern, erect quick tent-like walls to hide their work).
It's not easy, but after hearing about the Parshendi burning the occasional bridge (even if it was happening less and less frequently), and the fact that they're staying in the warcamps and taking forever to get to the gemhearts, I expected to at least see Soulcast bridges mentioned.
I think it's an effective method, but even if it was dismissed in the books, I'd be happy.