Bushrat,
Thanks for your observations—and your experiences with unsquared case mouths match mine.
The one tool I’ve found that helps straighten bullets without screwing up neck tension is the Tru-Angle, because it bends the neck slightly, instead of just pushing the bullet around inside the neck. But even then the run-out can’t be too bad before straightening. I’ll use it to correct a couple thousandths of run-out, but not much more.