I think effecting a change like this would require a massive cultural overhaul. It often seems to me as an outsider that Americans love debates and clear winners, which naturally leads to sophist arguments.
(In this respect it’s not much different to Australian culture, a good example being the ‘Q&A’ program which is just noise being thrown to and fro as a facade for discussion. The atheism-religion ‘discussion’ was just two audience groups booing and hissing slash clapping and whooping at each other’s figureheads. In contrast, because I can’t understand Swedish, their parliamentary debates seem fairly boring because they’re contained, calm and rational.)
This degrades debate from a way to intellectually understand the full topic (like a robust, live essay), to a kind of unintellectual theatresports, reliant on human emotion rather than reason. There always seems to be a fiery polarisation in every American political debate and sometimes I’m not convinced that there is an objective truth that is being argued by what Nietzsche called ‘inverse cripples’ (so expert in part of a field they’ve lost sight of the whole).
The debate system I think would contribute best (I can’t find the reference anywhere but it’s from an American institute) is for the opposing speaker to (a) summarise the previous speaker’s stance, AND (b) do so to the satisfaction of the previous speaker. This means maintaining respect and having to accurately frame your argument, not taking down straw men.