I do not think either the US or China has any meaningful role to play in the resolution: the Thais are and have always been extremely independent. They will trade with anyone, they will make security deals as it suits them, but they are not about to shape their own politics to suit anyone else.
This is true, but both China and U.S. will respond to the challenges and opportunities presented by this event to pursue their own interests.

I don't see the US being able to play the "China Threat" card to gain influence in Thailand, because the Thais don't see China as a threat
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I never said or came to close to implying that we could or should, that isn't the point of this discourse. The principle point is China will be more flexible in their response, while our options will be severely restricted by our laws, which could give China the ability to gain more influence with Thailand, while U.S. influence is reduced.

In the Philippines and Thailand you have what might be called pseudo-democracy, with regular electoral exercises offering a non-choice between or among elite factions with little or no ideological or policy distinction. That of course yields a great deal of cynicism and frustration and opens the door for charismatic pseudo-populist demagogues with no agenda beyond their own power and prosperity.
I think the Thai military would agree with you, their leaders have on more than one occasion called the Thai government a false democracy. I suspect they're more than a little frustrated with us blindly embracing the concept of democracy while ignoring the realities of how it is playing out.