I don’t think I’m being idiosyncratic at all. The OED definition you quote (the territory or land of a nation, usually an independent state) clearly observes a distinction between the territory (land), the nation (people) and the state (political construct), and makes it clear that the word “country” refers to the territory. From the definition a country is usually, but not necessarily, the subject of an independent state, but the terms are not synonymous; in 1776 Ireland would have been an obvious example of a country, and a nation, which did not have an associated independent state, and Austria an obvious example of an independent state comprising many countries and many nations.
Obviously in the case of, say, France, which had long been a distinct nation with a well-defined country and a well-established state the terms country, nation and state would be more or less interchangeable, but this wouldn’t have been true where nation-states had not been established. “Country”, when used of a country which was an independent state, could indeed refer to the state, as your quote shows. But that wasn’t its primary sense.
The Low Countries, for example, was a well-understood geographical and to some extent cultural term long before there was a political entity called “the Netherlands”. The United Provinces of the Netherlands, when established in 1579, based its name on the existing name of the existing country, the Low Countries.
The oldest sense of “country”, in English, refers to the land of a person’s birth or residence; the land with which he identifies. This could be what we might now think of as the national territory, or it could be a subdivision of that - even quite a small subdivision. In 1776, for example, General Charles Lee wrote to Washington complaining that “These Connecticutians [soldiers under his command] are more eager to go out of their Country than They are to return home”. In the context, it’s clear that their “country” is Connecticut. And this sense is still current; nobody who remembers the ads for “Marlboro Country” thinks that Marlboro Country was the United States. “Thomas Hardy country” is Dorset, not England. And of course England itself, although not a state, is unquestionably a country.
I think what the founding fathers - or at least some of them - would have felt is that by establishing the confederacy they were doing something which would establish such common bonds between the colonies as to make them not just several countries, but also one country, in a way that they had not been before. Or at least, it would give an impetus to the process. (The foundation of the United Provinces of the Netherlands might have been understood to have a similar effect, or the Act of Union between England and Scotland.) But this, to go back to the OP, wouldn’t have been seen by anybody as an occasion for making up a new name for America. America already had a name, and didn’t need renaming on this occasion any more that the Netherlands or Great Britain had needed renaming. The only think that needed to be named was the new confederacy, and “America” was the obvious referent for the new confederacy.