Why are religious studies scholars so vehement that they are not theologians?
There is a general definition of "scholarship" that specifies that, to qualify, a study has to be the product of research, using credible sources, the steps of which can be retraced by other scholars or academics. That is why textbooks, for example, tend to have many footnotes or endnotes: so that the reader can see what or who was the source of the author's information on a particular point or subject.
The Bible, as with other important religious texts or histories, is not scholarly in the sense conveyed in the above definition. While there are a great many members of the clergy who are scholars -- after all, most have attended academic institutions of higher learning -- not all scholars of religion are clergy. The distinction, as the question points out, is important. Many scholars of religious studies hold fast to that distinction because they believe fervently that attainment of knowledge comes from original or primary-source research. Theologians, in contrast, believe -- to greater or lesser degrees -- that the Bible, Quran, etc. is the word of God, and that faith in the word of God holds greater importance than research conducted by mere mortals. The whole point of faith, in other words, is that it cannot necessarily be scientifically proven, but it is true.