Essentially, my question is how accurate this picture is.
I know we wouldn't be that far ahead, but would there be any noticeable difference? I have always heard that christianity (namely monks in secluded monasteries) actually saved an awful lot of information that would have otherwise been lost by copying old manuscripts. Any light you can shed one way or the other would be very appreciated!
Well, it's completely false. One, I'm sure that they're referring to the entirety of the Middle Ages (600ish-1500ish). You can find a wealth of information in the FAQ section about the "Dark Ages." I won't provide a terribly in depth answer here as others have done a very good job of addressing this question: https://www.reddit.com/r/AskHistorians/wiki/darkages
Long story short though, monks not only preserved knowledge, but expanded it. It wasn't only monks, but other ecclesiastical structures, such as bishoprics also participated in this process. Numerous universities were founded by the Church in the Middle Ages.
