Think about the highest profile public trials in recent years - and I don't want to make this political of course - but thinking about OJ Simpson, the Clinton and Trump impeachments, the George Floyd murder trials, Rittenhouse, Depp/Heard - did the results of those trials change anyone's pre-disposed notions of guilt or innocence? We had a famous trial here many years ago involving Packers tight end Mark Chmura, a teenager and a hot tub. Chmura was acquitted (rightfully so), but if you ask around today most people here don't ever recall the verdict - they all just think he's a child rapist. Almost no one remembers that there were two defendants in that case - a second hottubber friend of Chmura's was also charged. Granted, he was not a famous person, but no one even knows his name or remembers his involvement because he reached a quick plea deal and did not go to trial (which was televised.) Very few people actually follow what goes on in these trials - they read a few (mostly inaccurate) reports here and there, and when its over they either cheer the result (I knew it! See??) or they decry the stupid judge and jury reaching a clearly wrong decision.
At the same time, a private settlement generally fades from memory with the passage of time. People remember there was that bit of trouble way back when, but if there's no trial it all remains safely hidden from view and eventually fades from memory. The optics of a public trial, especially if it gets televised, are what people remember in my opinion.