This conversation (to date) illustrates the problem we have with early Christianity. We just do not have enough information to be confident about the early developments.
To get too heated (as opposed to enthusiasm for an idea) will lead nowhere, because sooner or later you come to a dead end.
So it is with the GoT (abbreviation for the Gospel of Thomas) - I can't say that my exposure to biblical scholarship is all-embracing, but in the one year in which I attended weekly lectures and tutorials, (conducted by an excellent biblical scholar, with whom I got on well, in spite of his being a committed believer and knowing that I was an uncommitted disbeliever at a University with one of the best early Christian departments in Australia), I hope that I learnt something (Also note, I have no axe to grind, no doctrine to defend, just a passion to understand how I wasted my life in a unorthodox Christian group).
So which scholars believe GoT is Q? I can't remember any names being mentioned, although that's no proof.
And there is a another question for passionate believers in GoT being Q (or, at least a development of Q), do we then accept the Acts of Thomas as being true? If yes, then when you read, as an example, in the Acts of Thomas 57 (Apologies for the dated English, there are better versions available, but I've no time to search for one):
57 Again he took me and showed me a cave exceeding dark, breathing out a great stench, and many souls were looking out desiring to get somewhat of the air, but their keepers suffered them not to look forth. And he that was with me said: This is the prison of those souls which thou sawest: for when they have fulfilled their torments for that which each did, thereafter do others succeed them: and there be some that are wholly consumed and (some, Syr.) that are delivered over unto other torments. And they that kept the souls which were in the dark cave said unto the man that had taken me: Give her unto us that we may bring her in unto the rest until the time cometh for her to be delivered unto torment. But he answered them: I give her not unto you, for I fear him that delivered her to me: for I was not charged to leave her here, but I take her back with me until I shall receive order concerning her. And he took me and brought me unto another place wherein were men being sharply tormented (Syr. where men were). And he that was like unto thee took me and delivered me to thee, saying thus to thee: Take her, for she is one of the sheep that have gone astray. And I was taken by thee, and now am I before thee. I beseech thee, therefore, and supplicate that I may not depart unto those places of punishment which I have seen.
From "The Apocryphal New Testament" Translation and notes by M. R. James
Oxford: Clarendon Press, 1924
Clearly, these are all imaginative stories, told to influence the lives and actions of others. And, that is the function of all religious writings and also the oral stories, which are now lost to us.
At present Q is an imagined document. It may have existed, if it did, I imagine it to be like the hand-written notes that may be taken down at a lecture. Most likely, some people wrote down things they heard Jesus say and that impressed them, As his death receded into the past, and those who heard him speak got older, someone may have attempted to collect some of those notes. They would be scribbled of course, and brief. There were no desks to write on, when Jesus was lecturing. The sort of people we are dealing with were not scribes, and may have had only primitive writing materials, but if Q existed, it's origins may have been like that.