It wasn't Mohammed who founded Islam.

[pocaracas]

Really?
Right off the bat, the extraordinary claim that some allah made people have extraordinary memory so they could remember things from the far past without any contamination?
And, in spite of that, lots of similar sayings, but later deemed false, crept up...

Yes, the hadiths are not 100% reliable because it depended mainly on the memory of people, so falsehoods can creep up despite people having a good memory.

And it seems the main vessel for authenticity is the header of the hadith which contains the chain of people who remembered the thing, am I right?

Yes.

I'm getting the hang of this!

What kept people from attributing whatever they made up to some supposedly known person from the past? And how to discern who's truthful and who isn't?

Like your own Bayesian reasoning, if there were so many people attributing the same thing (i.e. Prophethood) to a single person named Muhammad, and if there is a high level of consistency between them regarding the issue, then most likely they are being truthful.

Depends on what their interest was when they made such claims...
I can't even begin to guess at it...
It also depends on when these claims were made. If they were made tens of years after Abd-Al-Malik's introduction of the madrassas and spreading of the religion, then... yeah... it is expected that many people will have the same basis, the same belief.
On that PDF, I saw many mentions of people getting information from a teacher... who also got his information from some other teacher... who got it from another... and it goes on and on... Teachers, I guess, imply a madrassa. So, we're talking about generations of teachers... too long for the claims to be reliable, no matter who is claimed to have uttered it before it came to the "committee" that was in charge of recording them all for posterity.

And about the rules this committee used, the way I see it, it seems that, at most, they would get internal consistency of all the hadiths...

Just like you have no hard evidence, I have no hard evidence either.

And don't you forget it!

So my lack of belief doesn't prevent me from accepting the possibility that some tribal leader attained, somehow (brilliant battle strategy, iron fist ruler, etc), a legendary status.

And what do you think that such a legendary leader was fighting for? Was it for the sake of political supremacy, religious, or both?

Errr... territory, land. That's the main usual suspect.
I'll just go ahead and take one more guess: maybe the land where his tribe was established became very poor and lacking in fauna and flora, perhaps the desert was creeping in... I don't know, I'd have to look it up. And they decided to take land from some other people who had decent land, decent food, decent animals.

Once you're the ruler of a good patch of land, good enough to give out some bits to your fellow fighters, you have to establish a political rule, in order to keep the newly conquered people in order...
Although, it seems that they mostly killed off all the men and kept the women and children(?), which would make it easier to keep people content.
Only a few years later would actual politics be required...

I'm trying to tell you that the prophetic role was probably added to the person by someone in between the legendary leader and Abd-Al-Malik.

So you accept one of the 3 possibilities:
1. Muhammad himself claimed to be a Prophet of God and he really is a Prophet of God (as all Muslims believe)
2. Muhammad himself claimed to be a Prophet of God but the claim is untrue.
3. Someone else added the Prophetic role to Muhammad.

I know you don't accept number 1 because you already said that you don't believe in the supernatural bits.

That leaves you now with either number 2 or 3 as a possible answer. But you said that you accept number 3 only. So, now you have to explain why number 3 is more likely to be true than number 2.
Why is it more likely that someone else attached the Prophetic role to Muhammad as opposed Muhammad himself attaching Prophethood to himself?

I'm not asking for evidence. Just asking for a Bayesian explanation to that, which you so love to use.

Well, yes, number 1 is too much to accept on mere say so... extraordinary claims require extraordinary evidence... I'm sure you've heard that, before.

Number 2 is possible, but the lack of contemporary writings and other evidence for it kind of lends little credence to it.

Why do I think number 3 is more likely? Because, the earliest evidence of such a claim comes many years... decades, even... after his "agreed upon date of" death. Also, remember that need for political ruling after having conquered the land... that seems to have come after his death, as well, with the caliphate.
In a way, it's a bit like the Teacher of Righteousness may have been the real person to whom the christ prophethood was later attached... some even go as far as claiming there were several such teachers of righteousness.