Why is uranium only in the crust, really?
(tl;dr below)

First, a correction. Siderophile elements are "iron-loving" elements, those that go in the core with the Fe-Ni liquid. Uranium is lithophile, or "rock-loving". It partitions to silicate rocky material (i.e. mantle and crust) relative to the core.

Secondly,


  But, according to this answer, the solubility of Uranium in the Earth
  Core is between 2ppm and 6ppm.


You have to understand where this is coming from. That answer is based on this paper, which experimentally measured how much U is in the Fe phase, coexisting with a silicate phase of peridotite composition (which is an approximation of the mantle composition). If you look at their Table 3 (page 14) and Figure 8 (page 17) you can see that the $D_\rm{U}$ values range from almost 0 to around 0.03. This value mean the ratio of U in the metal phase relative to the silicate phase. So even in the most extreme case, you're going to have 30 times more U in the mantle than in the core. And in their experimental setup they used wt % amounts of U in the starting material, much much more than exists in the Earth, so you eventually get a considerable amount in the Fe-metal phase. This does not mean that this is the amount that is present there now, inside the Earth.

The truth is, we do not know how much U is there in the core. And this research papers shows that whatever amount there is, there will be more in the mantle. There will be even more in the crust, because U is an incompatible element in partial melting of the mantle. See this Q&amp;A for more information about that:

What are the high field strength and large ion lithophile (HFS or HFSE &amp; LIL or LILE) elements?

Also related, also read the comments:

What percent of the Earth&#39;s core is uranium?


  Second, what I think: trying to mix unmixable liquids they we will
  have separate phases of solutions in the order of their density...


You have to differentiate solubility from partitioning. Solubility is how much of x (let's say U) you can put in y (let's say Fe) before you saturate a separate phase instead of one. Partitioning is when you have two phases x and y (let's say Fe-liquid and silicate), how much a third component z (let's say U) will dissolve in each. The paper linked to before was discussing partitioning. In terms of solubility, there is a complete solution of liquid U and Fe when the temperature is high enough (such as in the Earth's core). Even if there's not, a ppm amount of U in liquid Fe is not enough to saturate a separate liquid U phase, so density doesn't matter. If it's all one phase, density plays no role. If you have a solution of NaCl and KCl in water, it's all homogeneous. The K doesn't sink because it's denser.

tl;dr

We do not know how much U is in the core. We do know that whatever amount of U goes in the core, more U will be in the silicate mantle. When you melt the mantle to produce magmas that form the crust, it concentrates U even more, and depletes the mantle of U.