If the heavy metals are internal they are not extracted, as I guess you suspected from the way you framed the question. The resulting metals are often called 'total' but that is a misnomer.
Totals require that the clay minerals be dissolved, which requires HF or alkaline fusion.
I agree with Paul and Kees, the acid is simply being used to solubilize and dissolve the aluminosilicate clay matrix. The term of art used is " matrix digestion" and is performed prior to total metal analyses, to ensure a uniform, representative solution. Solids cannot be injected or introduced directly into AA or ICP instruments which are employed to identify and quantitate metals.
I agree with Carr, Paul and kees. I thing that your're wrong.
The acid mixture is being used to solubilice and dissolve aluminosilicate clay matrix and you can obtain a soil digestion and called "pseudototal content" (If can better that you use HCL+HNO3): If you want a "total content" you need HF or alkaline fusion, but HF digestion is very dangerous.
As Carr says, "Solids cannot be injected or introduced directly into AA or ICP instruments which are employed to identify and quantitate metals". If you want to measure solids such as clays... you need a measurement by XRF
I agree with Paul, Kees and Carr that all the acids can do,particularly the strong HF (handled with care) or alkaline fusion, is to dissolve the minerals in the clay into solution form the minerals in the clay. The dehydrated residue from the solution can be measured quantitatively using X-ray Fluorescence or XRD, using cavity filled or smeared thin section samples.
The acid mix can only dissolve the clay structure to release the minerals into solution (more effective with HF for total release) but cannot measure. Measurement can be then done using standard techniques.