I really think that it depends on the fabric! I have some fabrics that seem "stretchy" and I see it when I first iron it. If I can't iron it flat the same way twice, then that tells me that no matter how I work with this fabric, it's going to stretch! One argument for prewashing and ironing- it shows the character of that fabric (stable, stretchy, unravels easily, etc.) I have some fabric that this character has made me want to switch fabric before I ever cut it.
Other fabrics are more stable, and those it doesn't matter how you cut it. I have no idea if cutting on the grain exactly solves the problem with the fabrics that want to distort, because most of mine lately have triangles and you can't get triangles on the grain on all 3 sides. Starch really does help stabilize those fabrics, and I think that heavily starching those fabrics should help on a frame too!
Cutting on the grain may "help" with fabrics that tend to distort, but in my very limited experience that fabric will be tricky no matter what steps I do to tame it. Cutting on the grain and starching are two things people do to compensate for imperfect fabric.
The more stable the fabric, the less you need to do to tame it to get good results.