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I think we can find better words. Structural, maybe. Or even just: important.

Load-bearing feels like an unnecessary analogy.

Agree to disagree but I don’t think either of your words capture “the whole argument hinges on only one specific point. If this point is refuted then the whole argument is refuted. The rest of the points are only auxiliary but not crucial”

The word for that is "crux".

Also a great way to disagree collaboratively. If you can collaboratively find the crux of each others' arguments, you now know both know exactly what would change each others' minds, and you can find that information together and know what's correct.

“Load-bearing” doesn’t actually imply “only one specific point” (you can have several load-bearing walls). Anyway, you can use words like “critical”, “crucial” (as you used above), “essential”, “vital”, “key”.

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