Those languages to which people translate in the first step are called “hinge languages”. The machine people got the idea from the long-established practice, once they got around to speaking with some actual human translators (why are there so many people who think they can improve another profession’s processes, without ever asking the actual professionals how and why do they do things?).
A common language isn’t a sufficient condition to have a single country, obviously. But the question was whether it’s a necessary condition. It’s not that either, strictly speaking. However some sense of nationhood (the ‘human nation’ if it ever expanded to the whole world) is necessary for a successful nation. That comes out of a sense of a shared history and culture (including religion). For example in India there’s a shared history of that region being a quasi-single political entity, as under the British most recently before today’s India, who left English as the language of elites. The region that’s now India wasn’t always a political entity, but if it had never been, or the Hindu religion wasn’t common to most people in the various language groups, then India probably wouldn’t be a single country now either.
So all in all I think India or Switzerland (or Belgium, Canada or Spain as more fractious multilingual countries, but still single countries at least for now) are examples where shared history and culture can achieve some kind of ‘nationhood’ without a single language. OTOH the fractiousness in several of those cases, and the movement toward national separatism even where there is a common language (Scotland, the break up of Yugoslavia w/ multiple but in cases very similar languages, etc) tells you how remote is the possibility of a world nation with the current level of nationalism.
What usually happens in these cases is that the shared language follows from the shared political organization. That is, the language of the conquerors who unite the various areas becomes the prestige language of the polity, people learn that language as a second language for practical reasons, and their children or grandchildren might grow up bilingual in both the old language and the prestige language, and their children or grandchildren might grow up speaking only the new language. This can be purely a pull, where the language of the ruling class is emulated because of how useful it is, or there can be a push where use of the prestige language is mandated, education is only allowed in the prestige language and there’s a concerted effort to stamp out the substrate language.
This is what happened in Ireland. There are still a very few people who have Irish as their first language, but nobody in Ireland is monolingual in Irish. The same thing happened all over Europe to various vernacular languages that were replaced by the dialect of the metropole. Or not, there are still plenty of Italians who don’t speak “Italian”, they speak their local vernacular.
Anyway, if there’s a planetary political union, the people who organize this takeover will have a shared language, and this language will become the new planetary language, for as long as the planetary political union lasts. Even if the new elite doesn’t care if barbarian peasants in the hinterland can speak Planetary, the barbarian peasants might decide they need to speak Planetary so they can manipulate the levers of power.
The leader declares his or her own language to be the official language. Not everyone needs to know this language, but the government has a staff to translate official pronouncements into the various languages of the people. The deliberative body of the government (Congress or Parliament or the High Council, or whatever) carries on their business in the official language as well; fluency in this language (whether native or second-language) is a de facto requirement for becoming a member of said body.
The official language doesn’t even need to be a particularly common one. Look at how long the Catholic Church has gone with Latin as its official language, even though there aren’t any native Latin speakers any more.
The Austro-Hungarian Empire at the start of the 20th century: