Muslims, Bahai's, Christians, Some Hindus, and even a few Jews, quote sayings of Jesus as if they are historical sayings of Jesus himself. Of course most Christians would definitely believe the New Testament has his sayings in some form or another. Some believe they are absolutely verbatim, while some believe it is the inspiration worded by a human being.
Muslims typically use the New Testament quotes to validate their own faith. Bahai's do the same thing. Christians of course as understandable would use all of it for their whole theology or more. Some Hindus who believe in a Bahai like theology where a new representative of God is the incarnation of Jesus himself would use the New Testament to derive some his Jesus's quotes for their theology.
I cant list all the institutions who do this so please understand.
Other than the methodology of "faith", what other historical method do you use to validate any of Jesus's attributed statements in the NT?
I used various means before I abandoned all these methods together. One was multiple attestation, if two independent sources which didn't know another reported a saying then I thought it was likely He said it (likely is as best as this area can get from all I read). One which I read about was if the saying was in Aramaic or how it was translated showed that it was translated from Aramaic, and "Jesusisms" of language like, "amen amen I say..." and so on. Another was that the historical Jesus was considered to have likely told parables and not allegories, so if something was allegorical in the Gospels it was likely edited on to an original saying. There is some evidence for this if you consider the Gospel of St. Mark and the Gospel of Thomas, the parables told in those differ from later writings, so I would often consider the ending punch-lines allegorizing the parable something that Jesus didn't say but were edited on to what He said. This also would mean that the speeches in the Gospel of St. John were not authentic to what He said (as it is utterly different from how He spoke otherwise). Similarly when a parable differed in it's telling I'd consider the earlier writings to be more accurate to what He actually said rather than the later ones. If something would have been considered embarrassing in the culture, or dissimilar to currents before and after (Jesus not fasting for one), I'd also consider that as likely authentic to what the historical Jesus said. Considering His social class also assisted in this.
But as I said, the best this whole area gets is "likely." They were things I was learned about that groups such as the Jesus Seminar, Context Group of scholars, and others applied (for I liked to learn how to make historical arguments myself rather than cite a book, it is a pet peeve of mine when people just cite a consensus and do not make the argument themselves, although I am probably hypocritical here at times).
The end result was a rather strange picture of a country guy who was concerned with the ethics about money (fits the social situation in the area at the time with people getting their land taken for various reasons and increasing poverty), an imminent Kingdom of God which would restore the positions of everyone, who went around healing people (not an uncommon thing but I have no idea what this would've involved or been read as in the culture, but the practice of spiritual healers is not uncommon anywhere), and didn't keep the Law very well.
That's all in my opinion and what I was using and learning about before at least. It works by collecting the sayings and actions with their context and applying those methods to each of them.