I think you would be hard-pressed to argue that the Department of State, and its counterparts in Canada, the UK, France, Germany, Spain, Italy, Greece, Brazil, Japan[1], etc, etc, etc, have been 'highly leveraged by China for political gain'.
Oh, and to throw another monkey wrench in your argument, consider that both China and Taiwan believe that Taiwan is not an independent country, but that it is part of China.
What they disagree on is who is the legitimate government of China.
The question of whether or not ISIS is therefore a country is left as an exercise for the reader.
Both governments make claims against the other's land but that doesn't make them the same government.
Edit: if you look at the font in question, it does have flags for England and Aruba, for example, so offering the flags doesn't imply that you are recognizing the places as sovereign states.
Compelling other people to do things, with either a carrot, or a stick, is an incredibly normal state of affairs in life. My landlord will have me out on my ass if I don't cut him a check on the first of every month - does that mean that my recognition of his authority to my apartment is illegitimate, or somehow coercive?
2. Is Taiwan a distinct nation, or is it part of the Chinese Nation?
Both the PROC and the ROC currently seem to think that it is the latter. Most of the world agrees with them.
(Bonus points: Is Catalonia a country? What about Cascadia? What about Transnistria, and South Ossetia? What about Crimea? By your definition, it seems to quite clearly be part of Russia... Be careful where you express that viewpoint, though, it's not one shared by most of the world's governments, or most Ukranians...)
Anyone in these departments of state is aware that Taiwan is a separate country. They’re just not able to say it.
The larger area you call the Netherlands that includes Belgium and Luxembourg is currently called the Benelux[1]