Why was it important what the public thought?

Because the public already loves Amy for being the inspiration of the books they grew up with, and the media only fueled their outrage for what they believed happened to her, and that would have influenced the outcome of Nick’s trial. As Tanner Bolt rightly pointed out, it would have been a huge re-alignment.

This was a very public case. Any jury that would have tried Nick would know about it before being impaneled.

Going to the talk show meant he could change the narrative and influence future jurors.

Because if the public thinks that you got away with murder, your career, personal life, and family life are fucked. You become unemployable, undateable, and your family wants nothing to do with you, and you're going to be harassed by the police and sued by anyone who thinks they might have cause, because they all think you need to be punished.

See O.J. Simpson for an unusually well-off example. He'd be shooting up fentanyl under a bridge somewhere, if he didn't have an NFL pension that can't be touched by lawsuits.