If you really want to pursue this discussion, there are MANY posts about it from past years and I personally find the discussion a bit numbing. I honestly think the only people who believe Barnard is a back door into Columbia are people on sites like this. That isn’t a sentiment you often hear at Columbia. And as for Barnard students not having a “right” to a Columbia diploma, that’s odd as all Barnard degrees are conferred by Columbia as it is Barnard College of Columbia University.
I wouldn’t base a decision on what you hear or on questions like this. Better you decide whether you want to attend a Women’s Liberal Arts college in NYC and you want to be part of a major university.