I'm not sure exactly what you have in mind when you say 'naieve direct inverse'.
In general, EViews uses Cholesky decomposition of moment matrices to solve least squares problems. There's no inversion involved, because you solve the triangular systems directly by forward and back substitution.
The decomposition algorithm used doesn't really matter much here (Cholesky vs QR vs SVD). The real issue is whether we factorize the original data matrix (X) or the moment matrix (X'X).
We generally work with moment matrices because the calculations are faster and require less memory. The downside is that we may report singularity errors on severely colinear data in cases where a QR or SVD decomposition of the original data matrix may have produced results.
There's some subtle issues going on here, but for most real world statistical problems, it's unlikely that you would see much of a difference in results.
There are currently no options provided to change the method for least squares, although it would be possible to write an EViews program that did simple least squares calculations using the SVD of the data matrix.