That's true of the sort of debate that's practiced competitively by "debating teams", but I don't think it's always true of debate as the word is used in an everyday context. A debate can also be more like a dialectic.
That's the problem with how we see a debate. It's just about that. There's no need to present your position in a compelling way or find common grounds with whoever you're debating.
If you read any books by professional negotiators - you'll notice that they conduct themselves very differently than you would in a debate.
Getting to Yes and Getting Past No have been eye opening to me. I no longer feel like I even need to "win an argument".
> the public can gain a lot because a debate forces you to be very on point
I recall reading some articles stating, that debates fail to convince anyone of anything. Public debates only encourage tribalism, IMO. If you watch a presidential debate - no amount of "winning" changed people's opinions.
I can even argue, that just hearing "X won the debate" will cause more impact; compared to listening to them.
What people often don't realize is that winning a debate doesn't necessarily mean you side with the truth. A debate is aimed at winning on your existing opinions and the other losing. It is not aimed at discovery, validation or learning. Victory is more important than truth, and a lot of 'good moves' in a debate actually bring you further away from the truth.
If you engage in a conversation where you are both curious and submit to what's true (whatever that means), this conversation will rarely be a debate.
If a debate is in public between skilled debaters who show sportsmanship, then I think the public can gain a lot because a debate forces you to be very on point.