For obvious reasons, if you're in a French language group for non-native speakers, the word 'speaker' comes up quite a lot. I've asked two native speakers the question asked in the headline and they both said emphatically no, and one verified it with her husband. Yet this guy at 4:15 of this video uses the word. Then this graph shows that 'locuteurs' is used roughly a third of the time as 'speakers' in English at least since 1980. This website shows five uses of the word, though I'm aware that there are some machine translations or bad translations on the site. So let me know what you think.
Of course locuteur / locutrice du français is a correct translation of "speaker of French". But it is not the only one. What to say comes down to register. I will try to address all the points in your question but in a different order.
The ngram in your question shows that the word locuteur wasn't used in French before the 20th century whereas English has used the word "speaker" since at least the 14th century (OED gives 1303 for the 1st occurrence). This of course doesn't mean that the concept of speaker didn't exist in French before that time.
Locuteur was introduced in French by Damourette (a grammarian & linguist) and Pichon (a grammarian & psychoanalyst) in their monumental1 work: Des mots à la pensée, essai de Grammaire de la langue française, published between 1911 and 1927. In the glossary they define the word as personne qui parle. The word is first used in vol. 1, § 48. Damourette et Pichon have been remembered by linguists as the "inventors" of the concept of locuteur2.
This explains that the French word locuteur is used in linguistic contexts. Using it in everyday speech shows a certain degree of schooling. I would say locuteur is used in French when the word "locutor" can be used in place of "speaker" in English3. "Speaker" has also much broader uses than locuteur since it is not restricted to linguistic uses, hence the relative non significance of the ngram shown in your question.
The video you link to in your question is made by a linguist, so the use of the word locuteur is expected in this context. Most of the time, in everyday casual conversation in French, the "speaker" is celui qui parle. Celui (celle, ceux) qui parle français, celui qui parle anglais etc... sometimes there are specific words (anglophone, francophone etc...) as said in another answer. But these specific words would not be used under any circumstance, here again we have to consider context and register. Most of the times if I want to say that, for example, someone is a good English speaker, I'd say il/elle parle bien anglais, because being francophone or anglophone or whatever -phone, situates the person in a community, it's not just a casual remark about themself.
Where I will say "I'm a French native speaker" to an English person I've just met, I would say ma langue maternelle est le français in a similar situation to a French person, but here on FL saying je suis locuteur natif du français is not a problem because I know I won't make people uncomfortable using a linguistic register. When studying a literary work in an educational context locuteur is also the right choice of word.
If talking about the speaker on a stage, at a meeting, etc... I'd use the word oratrice / orateur.
1 7 volumes + table of contents and glossary.
2 La grammaire nationale selon Damourette et Pichon : l'invention du locuteur, a thesis by Valelia Muni Toke.
3 The wiktionary says that "[Locutor] was very rare until the mid-1900s, and is still less than a thousandth as common as speaker.
To improve what LPH said, there is one other translation to speakers. We will often use the -phone suffix, for example :
- French speakers -> les francophones
- English speakers -> les anglophones
- German speakers -> les germanophones (even though German is allemand in French)
- Chinese speakers -> les sinophones
And we will use -phonie to describe every speaker from the language, e.g., la francophonie is the community of French speakers around the world.
I can't understand that the fact you find three times "speaker" in English for each time you find "locuteur" in French proves something about the translation of "speaker". Be that as it may, one can say that there is no doubt that "locuteur" is a proper translation, if not the translation of "speaker" in such expressions as "speaker of French" and "speaker of Basque". Here the ngram for "locuteur du français".
The Google page of examples corresponding to this graph shows numerous cases of use in works dedicated to teaching, the art of language, and certain matters in sociology and other academic disciplines: "examples of use of "locuteur du français"
You can also find "locuteur de français", but it is much less used although found in similar books. However, you can use this option if you feel like it, as it must be reckoned with as correct. Let's notice however that "locuteur d'anglais" is not used, which fact, again, is not tantamount to saying that it is not an allowed expression.
Personally, I see no other translations. There is "utilisateur du français", but it is muxh less used and the meaning is not quite the same: "utilisateurs are necessarily "locuteurs" but not vice versa.