Trade in War

It's simple self interest. If you perceive trade as benefiting your interests (which, by definition, would imply not giving your opponent a meaningful edge in the conflict) then you trade.

It's the same thing that decides when wars end. Either side is free to surrender or withdraw at any point in a war. They continue fighting when they think that the result they can obtain (win or lose) in the future will be better than the result they can obtain in the present. They stop fighting when that no longer becomes the case, at least assuming non-insane leadership.

War isn't the end of relations. It's not infrequent that enemies today are allies tomorrow, or vice versa. Nobody ever knows what tomorrow holds, and history seems to love a nice plot twist.

Americans mental assessment of war is that it is an all out slog, you gear up the entire country and fight to the death. Of course, that image is changing as the US continues to engage in protracted conflicts in ponzi states like Afghanistan, Korea and so forth.

However - for older nations, especially european ones, war is another tool in the chest of diplomacy, and may not be viewed as 'all encompassing'. To understand that war is just another lever of power can help Americans make better sense of conflicts.

You're ignoring the principal-agent problem. Even when national leaders aren't technically insane, they will often choose to continue fighting well past the point where it could possibly benefit their country because that suits their own personal interests. So they keep sending their citizens to die because there are no personal consequences.

The GP did not say anything about war continuing due to logical outcomes or nationally-motivated ones. Most wars simply are personal pursuits wrapped in convenient excuses.