That's like saying spoons are more flexible than forks because you have soup (rotation matrices). Spoons and forks both work for rice (pos def matrices), and you'll want a fork for noodles (rectangular matrices).
The SVD staying in Reals when you have Real data is a nice feature.
This is exactly why eigenvalues are less flexible than singular values. You can have a real valued matrix that does not have any eigenvalues in the field of real numbers, all eigenvalues are complex numbers. Examples: rotation matrixes in R^2 have no eigenvalues in R. Singular values, on the other hand, are always real (they are eigenvalues of the Hermitian matrix MM^*) and can be used the same way for the matrices over real (R) or complex (C) numbers, hence extra flexibility. Added bonus -- singular values are never negative.
Instead of "more flexible" a better statement would be -- eigenvalues convey more information than singular values.