It wasn't Mohammed who founded Islam.

[Rayaan]

That it passed down intact and unchanged is a "mighty if", yes.

There may be sporadic variations here and there, yes, but large systematic variations are much more unlikely to occur, especially if there is a common and very basic idea that exists in at least most of the oral accounts. The idea that Muhammad claimed himself to be a prophet is something very big (and recurrent) in the oral traditions, so it's equally unlikely that someone else falsely attributed this claim of Prophethood to him at a later time without it being noticed by anyone else during that time.

I've been reading an interesting book on this subject, titled Memory in Oral Traditions: The Cognitive Psychology of Epic, Ballads, and Counting-out Rhymes. In the book, the author has carefully presented a comprehensive model - by synthesizing a multitude of empirical, theoretical, and historical research - of how oral traditions passed down through memory is more reliable than people nowadays imagine it to be. One of the conclusions is that "Oral traditions maximize memorability so that information can be stored without external memory aids for long periods of time" (p. 317). Evidently, there are many examples of ballad verses, poems, and songs produced by oral cultures in which the same basic ideas and some of the poetic structures have remained stable for centuries. The same holds for the preservation of everyday human affairs as well:

"In many situations, oral traditions provide a more appropriate model of everyday human behavior than do psychological experiments on memory. At times, people do have to remember what exactly happened on one particular occasion, as in eyewitness testimony. It is much more common, however, to abstract and remember the structure from many similar events, no one of which by itself is the best version. In general, people are better suited for the more common task (Bartlett, 1932; Neisser, 1981; Rubin, 1986)." - David C. Rubin, Memory in Oral Traditions, p. 7

"Western discourse has come to prioritize the written word as the dominant form of record keeping and until recently, Westerners have generally considered oral societies to be peoples without history. This could not be further from the truth. Oral societies record and document their histories in complex and sophisticated ways, including performative practices such as dancing and drumming. Although most oral societies, Aboriginal or otherwise, have now adopted the written word as a tool for documentation, expression and communication, many still depend on oral traditions and greatly value the oral transmission of knowledge as an intrinsic aspect of their cultures and societies."
http://indigenousfoundations.arts.ubc.ca...tions.html

"The western emphasis on textual evidence indicates an ethnocentric aversion to non-western methodology. Cultural blinders exist in many forms of historical research, like the written word, which limits the amount of knowledge a researcher can retrieve from a document. Certainly the same sort of limitations apply to different forms of textual research."
http://www.nps.gov/subjects/ugrr/discove...d-myth.htm

And how do we come to know about those muslims?
Could the source that tells us this history be skewed? Could it be exaggerating? Could it be biased?
Could it be just boosting its numbers?
Could it have found the source of the rumor?

I find those questions pretty ironic given that you yourself were so conveniently bolstering your own rumor hypothesis with weak, non-verifiable responses such as "mighty IF," "It's a possible scenario," "That is one option of how things came to be," "This is another potential way things could have unraveled, for all I know" ... and yet now you're asking me questions which I cannot possibly verify with evidence. That's a nice double-standard you have there.

You don't accept oral tradition as history, so there is no other answer to those questions that will convince you, it seems.