Performative is extensively used by government officials too...
You continue to list examples rather than actually provide statistics, but you speak of "frequency". That isn't a good argument. Just imagine if my response here was to list all of the times people use imperatives: "Bosses at work also use imperatives!" and so forth. No one is denying that this usage exists or that it is associated with certain contexts.
IMHO, it’s a pretty common speech mood in certain contexts.
Yes. But "pretty common... in certain contexts" doesn't (necessarily) make it relatively frequent overall in comparison to other aspects of language. And the only way we can really discuss that is with some general measure of frequency, not anecdotes/examples.
A judge says “You are convicted to a 10 years sentence.”
A member of the administration signs orders that have practical effects.
The problem with these examples is that even in the way they are expressed in English, they are ambiguous. Yes, there is some effect of performance in addition to what is stated, but these are often just declaratives ("you are convicted") with legal (or other) force. This is another reason that there might not be enough pressure to grammaticalize: they can be expressed in another way.
Performative is missing. ;-)
Definitions typically give only the most salient examples, and they end up a bit circular, based on what is considered central in previous definitions or is researched the most. You've made a case that performatives could receive more attention, and I don't think anyone disagrees. But in order for that to happen, someone, a researcher, will have to take it on as a project, and I've suggested some ways you could do that. And Vox made some points about how these things actually have been studied but under different names, so that's another starting point, to look at that previous research.
Yet initially my question was not why a grammatical performative mood doesn’t exist in the languages I know, French (besides “Que” + subjunctive) and English (besides “Let” + infinitive), but whether some languages might have one.
1) What is "a mood"? English "let" is a specific construction that might fit your definition. But are you only looking for verb suffixes or something like that?
2) I offered the potential example of Arabic. Vox had some other suggestions.
Similarly performative is expressed by a variety of grammatical moods and the context tells what’s actually meant.
As above, I would be cautious to distinguish between whether a form
expresses performatives, or whether certain forms are used performatively. Declaratives, as I said, can be used that way, but that doesn't make them necessarily special morphosyntactically.
As a matter of fact I suppose that many people wouldn’t consciously admit they are trying to perform something just by speech.
Perform what? Someone leading an official ceremony or legal proceeding, etc., would certainly be aware of this. And any time when the speaker is not aware is arguably less clearly an example of a performative. I'm not really sure what you're trying to argue here. Additionally, all language is performative in a sense-- if I ask a question, I'm sort of giving a command for you to answer the question, and if I give a command, I'm sort of proclaiming that you must do what I say, and so forth. Or even with simple declaratives, I am asserting that you should believe my description of the world. Many non-linguists would not be aware of that explicitly, but it is trivially true and not hard to convince someone once you explain it.
So as with other aspects of this discussion, it will be very helpful if you provide specific definitions (especially operationalizable definitions, e.g., that you can apply as diagnostic criteria), and also some quantitative data on frequency or other relevant points.
A corpus study consists in studying actual speech acts, right?
A corpus is a body of text. Corpus studies look at that text and report on the distribution of linguistic forms. The majority of that research involves some sort of frequencies. So you identify things, then you count them, then you compare them. There are very complex (e.g., mathematically or computationally) methods, but they almost always boil down to that basic idea: what are the relative frequencies of X and Y?
Actually I would say that performative is used in specific circumstances where the utterer either has an actual power (government officials, Yahweh, Jupiter), believes he has (in religion, magics, etc.) or wishes he has (sports fans, cheer leaders, etc.). I guess it must be mainly oral.
The entire point of corpus research is to get away from speculation and intuitions. What I've suggested is that you take a corpus and annotate each utterance for speech act. Many will be declaratives, some will be interrogatives, etc., and then you will also find some performatives. Then you can report a simple distribution, where 50% are declaratives (or whatever), and finally something like 5% are performatives. Then we can actually talk about this in an objective way.
The very fact that performatives seem to be strongly associated with specific discourse contexts is a strong indication of why the language may not have a generally grammaticalized form for them. (And on the other hand, why specific contexts do have specific traditions, such as "I now pronounce you man and wife...")
I was not thinking about the evolution of idioms (langues in French) but of language in general (langage in French).
Well, we can't observe that. Almost everyone would agree that language is no longer evolving in that sense, and hasn't for thousands of years. Most linguists (not all, but most) today would not claim that for example the earliest written language is any evidence about an early 'less evolved' form of language. The fact is that all humans share the same genetic capacity for language (and that's tens of thousands of years or more). That is one reason research about the topic is controversial: we have no direct evidence.
But you can read a lot about it in current research. So rather than reinventing the wheel, you should start with at least some of that.
There is no trace of animal language remaining today in ours.
Actually some recent research argues there is. Some approaches talk about "syntactic fossils" or basic constructions that might hint at an earlier proto-language. Controversial. But it's out there. Again, do a literature review if you want to know more about this.
La Société [Linguistique de Paris]...
Yes, because there was at the time too much speculation about the topic and it wasn't going anywhere. But that was just one group, and there are many books written in the past 5 or 10 years on the topic for example. It wasn't really taboo then either, just overdiscussed with no clear advancement from continued discussion.