Sorry, no. You are not right, and the sentence is correct. At the point where this sentence presumably appears in a text, we have been told two things. We have already been told that some languages are about to disappear, and we have also been told about the existence of some minority languages. Then, with an implied 'However..', the sentence tells us that - contrary to what you might suppose - it is not these languages which are most likely to disappear. The sentence which follows this one will tell us that it is, in fact, certain other languages which are going to disappear. The languages with the smallest number of speakers are not necessarily the ones that are most in danger. The word 'necessarily' is not the same as 'necessary'. 'Necessarily' means 'It doesn't go without saying that this is the case/Don't automatically assume that this is so.' By the way, you can always rely on the British Council. There is a lot of poor English around these days, especially on the internet, but you can still trust the British Council to get things right!

The original sentence is correct. Here necessarily is used to talk about inevitability, in this case, the inevitable result of languages disappearing. It is saying that just because the languages are not widely spoken, that doesn't mean they are the ones that will automatically disappear. It is not inevitable that these small languages are about to disappear.

I believe this kind of sentence is sometimes called an "emphatic sentence", but this is not a widely established term. An emphatic sentences emphasizes specific part of a sentence, as in this example. 1. I love dogs. - Plain. 2A. It is I who love dogs. - Emphatic on "I". 2B. It is dogs that I love. - Emphatic on "dogs". Now, if we transform "These small languages are not about to disappear." to emphasize the subject, we get: "It is not these small languages that are about to disappear" We add "necessarily" after "not" to make it qualified, like "even though there might be rare exceptions", for example. 3. It is not necessarily these small languages that are about to disappear. Now consider what you suggested. 4. "It is not necessarily that these small languages are about to disappear." This is also a correct sentence and may be called emphatic too, but obviously it doesn't have the same meaning as #3. #3 emphasizes a part of the sentence, while #4 emphasizes that the whole hypothesis is not the point of the argument. ("It is (not) that ..." means essentially "It is (not) my point that ...") If we want to express doubt on the veracity of the hypothesis, we would rather say this: It is not necessarily true [certain / a given / a foregone conclusion] that these small languages are about to disappear. So we can say the #3 and #4 sentence you were wondering about have different meanings and different emphases.

The sentence is fine. It is not necessarily the previously mentioned minor languages that are about to disappear.

Thanks.