This is indeed the goal of formal debates, which is why debaters are trained to argue both sides of an argument, and the winner is judged by who the audience found most convincing, rather than by drawing concession from one's opponent.

The goal of rhetoric is not to make your opponent admit defeat, which is why you don't see politicians try to actually engage with an opponent's argument.

Likewise, a lawyer arguing in court is not trying to convince the other lawyer.