You seem to be speculating based on unfounded assumptions. Luckily we don't have to assume or speculate, because we know quite a bit about the interior of the giant planets these days.
> a mind-boggling amount of rocky and metallic meteors.
With a total mass a mind-bogglingly tiny fraction of their original rocky-icy cores. Just like with Earth, any post-formation mass increase is a rounding error of a rounding error, because by definition the era of planet formation stopped when the planets ran out of raw material to capture!
> They have to go somewhere, and it's unlikely they're just dissolved into the atmosphere.
Unlikely by what argument? Essentially anything that falls into one of the giant planets is vaporized high in the atmosphere, long before even hitting the cloud tops! Some of the gaseous meteor stuff may recondense into microscopic dust particles that will remain in the atmosphere due to buoyancy and other forces that vastly exceed gravity at those scales.
Any hypothetical chunk of solid matter that somehow falls through the upper atmosphere and the cloud layers intact will encounter a layer of hot supercritical fluid at tremendous pressures, thousands of km thick, and get quickly dissolved.
And below the supercritical fluid there's an even more exotic mantle that comprises around 75% of the total mass of the planet and most of its volume. The mantle is made of extremely hot, extremely reactive, exotic liquid metallic hydrogen on observations of the giant planets' magnetic fields. There's absolutely nothing that can fall through that intact even in theory.
> So there must be some amount of molten rock and metal at the core
No molten rock or metal can exist in the core due to the pressure. However, as I said, there are definitely several Earth masses worth of silicates, and some but not much metal, in the core, because that's the original core that formed first and started attracting gas. Again, the question is whether at least some of that material forms a well-defined solid core, or whether all of it has been dissolved away by the metallic hydrogen mantle surrounding it. The best current models indicate that there's no well-defined boundary and the entire core is "fuzzy" and mixed with the mantle.