A very insightful question. The answer is that, yes, to some extent there is a kind of circular reasoning involved, but it is a bit complex. This is similar to how fossils are dated. Paleontologists use index fossils to date fossil-containing rocks, and they use uranium/thorium/lead dating to measure the age of the same rocks. The circular reasoning is that these scientists sometimes use the presumed age of a rock from the fossil it contains to check the accuracy of the uranium/thorium/lead dating, but at other times the same scientists use uranium/thorium/lead dating to determine the age of the index fossils. This probably sounds pretty “fishy” to non-scientists, and not completely without reason. What is really going on is that scientists have a whole range of both index fossil and uranium dating data. Both are checks on each other. In other words, up to a point, the index fossils can help to confirm uranium dating when there is some doubt about the uranium isotopic dating and up to a point, the uranium dating can help to confirm the index fossils when there is a question about the fossils. When the two are highly consistent, then the the scientists are much more confident of the conclusion. In this system, if either uranium or index fossil data is discrepant, the overall combination of evidence from both methods can be used to try to decide which is problematic–the uranium or the fossil evidence. The two combined provide more, not less accuracy and believability.
The situation is similar with archaeology as it is with paleontology. There are multiple ways to attempt to measure the ages of things. Some of them are 1. C-14 dating of organic matter 2. Things which change over time in fairly predictable ways, such as pottery style, writing script style and other cultural items. 3. Stratigraphy–using the location in a succession of layers to bracket the age of items. If you are below a layer which is for sure about 1300 BC, then you are older than 1300 BC. None of these methods is fully reliable or perfectly precise. But… there are an extremely small number of absolute checks on ages that can be used. For example, on rare occasions, Babylonian or Mayan sources may record a solar eclipse. Even more rarely, there have been a couple of extremely massive volcanic/earthquake events, such as the Santorini eruption for which we have a fairly precise date. When eclipses and eruptions can be cross-checked to archaeological data, this provides another way of checking dates.
So, let me get back to your question. The answer is that you are right, we tend to use other dating methods such as C-14 dating to estimate dates for certain types of manuscript, and therefore to try to estimate the dates that certain scripts styles were used. But then we use these script styles to try to date other items. To some extent, these different methodologies of dating objects are used in a reciprocal way (which is a nice way of saying there is some circular reasoning going on). However, the different methods of estimating the ages of things have a way of cross-checking one another, and this is a good thing, not a bad thing. What careful archaeologists do is they try to use as many of the methods in concert with each other as possible. When script style agrees with C-14, which also agrees with stratigraphic data, which also agrees with cross-checked historical data, then the archaeologist begins to have really good confidence in the proposed date. Manuscripts are tougher than things like pottery because stratigraphic data is not very helpful. People tend to hold on to really old manuscripts. If we find a manuscript in a layer of a dig which is from 750 AD, the manuscript we find there can possibly be much older. Pottery usually has a much shorter half-life of usage. So, you should look at the proposed dates of manuscripts, but also try to judge for yourself the quality of the data being used to estimate the age.