I think it sometimes depends on how much they have internalised their perspective on a topic as a core part of their personality.
If they perceive a disagreement with their perspective as a direct attack on their person, that can lead to subjectively bad outcomes.
There is also the possibility that what you see as a small point is a critical point to them.
Perhaps I'm playing in to the scenario OP is describing but I'd argue that being wrong (let's assume for this example it's provably, objectively wrong) isn't necessarily weakness, sometimes it's just incorrectness.
i'm possibly drawing a pedantic line between weakness (a potentially valid, but weaker argument) vs incorrectness ( an argument that is provably, objectively incorrect ).
Perhaps i'm just describing the difference between subjective and objective arguments ... hmmm, not sure