I disagree on two points.

1. You mis-characterize the challenge of how the public should react to character assassination attempts. This is clear in the contradiction between “don’t support character assassinations at all” and “evaluate ideas on their own merits, not based on who’s bringing them up.” Sometimes, evaluating the merits of ideas brought up by attempted character assassination will lead us to effectively support character assassinations.
2. You mis-characterize the dynamic of Carlson critics. They’re not faking their concern. They were concerned already, which means new evidence of what concerned them enforces their concern. They wanted Carlson gone before hearing his old comments, and now they’re even more convinced they’re right to want him gone. You’re treating opportunism and conviction as opposites, but they often go together.