In response to your second point, those are not horseradish, on which we have a mesorah, so we work backwards, as the Gemara did with the identification of Pri Eitz Hadar as an esrog.
Which brings us back to your first point; when there’s a legitimate mesorah on a mehalach in Yiddishkeit, no amount of debating it will make it wrong, and when there us a mesorah to reject an approach, no amount of debating will make it right.
There’s another important point as well; not everything is subject to logical debate. If I observe that a particular approach simply doesn’t work, or is not grounded in sincerity, it may not be provable on paper, but is no less true.
Let me take the women wearing tefillin issue, on which we agree, as an example. I think I’ve demonstrated that according to classic halachic sources and approach, it is assur.
But Rabbi Schachter took it a step further, and said that it is unacceptable for “meta-halachic” reasons. So the logical, halachic debate doesn’t really need to begin. And, at least according to my reading of Rabbi Twerski’s piece, we also take into account the motivation of those involved in the innovation (even while judging favorably the motivations of the SAR students).