I think your opinion makes a lot of sense in principle, but doesn't actually work out in reality. Suppose DPP separatists did something that finally crossed China's red lines. Assuming the US quickly made private assurances, e.g., Milley called and pinky swore over the phone, Xi then called Biden to confirm.This isn't about China caring about public opinion, it's about the US caring about global public opinion enough to reduce the likelihood of abrogating its word by a small amount, compared to the likelihood of abrogating its word if they went for a private assurance only option.
Again, I think both "private only" assurance and "private+public" assurances are both not super useful or iron clad if the goal is to try and prevent the US from intervening, but I rate "private+public" to be marginally more useful than "private only".
But at the end of the day it is likely that if either option is chosen, the US will still try to do its best to support Taiwan as much as it can during a conflict, to such a degree that China may view it as crossing a red line.
I don't have anything else to say on the matter of "private" versus "private+public".
What do you think the US would publicly announce before and after AR?
Option A)
Before AR: Taiwan has some bad folks also, therefore they are unworthy of our spilled blood.
After AR: See, we told you so, all of Taiwan is infiltrated with CCP sympathizers.
Option B)
Before AR: Taiwan is our strong ally. We will defend them at all costs, but Xi reassured me China is not launching a war anytime soon.
After AR: Those damn lying commies! Fool me once, shame on you.
One of these options is clearly politically untenable, in fact if the US made public assurances China would get even more suspicious, since it would be choosing to pay a high political cost for minimum benefits and it would go against the previous US traditions in times of crisis.
I believe this is why logic of additivity doesn't necessary apply here, adding public assurances might not be better than purely private assurances.
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