And you can use a spoon to chop spaghetti into pieces. Say I have a 10,000 by 100 matrix (1M elements), squaring that to get a 10,000 x 10,000 (100M elements) matrix to get the singular vectors is probably a bad idea. That squares the condition number too.
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.
The SVD staying in Reals when you have Real data is a nice feature.