Why can't the union of these regions be displayed completely?

Let $Omega$ be a spatial region bounded by cone $z=sqrt{x^{2}+y^{2}}$ and hemispherical surface $z=sqrt{R^{2}-x^{2}-y^{2}}$, and $Sigma$ be the outer side of the entire boundary of the region $Omega$. Now I need to find the value of second-kind surface integral $iint_{Sigma} x d y d z+y d z d x+z d x d y$.

Region[reg = 
  RegionUnion[
   ImplicitRegion[
    z == Sqrt[x^2 + y^2] && Sqrt[2^2 - x^2 - y^2] >= z, {x, y, z}], 
   ImplicitRegion[z == 2, {x, y, z}]], 
 PlotRange -> {{-2, 2}, {-2, 2}, {-2, 2}}]

But I found that the union of the above regions cannot be displayed in all, but each individual region can be displayed. What can I do to correctly display the union of these regions?

   surface = z == Sqrt[x^2 + y^2];
Integrate[{x, y, z}.(Normalize[D[surface, {{x, y, z}}]]) /. 
  Abs -> RealAbs, {x, y, z} ∈ reg]

So I can’t directly solve the above surface integral. What can I do to solve this surface integral correctly (without the help of Gauss formula)?