It is not selection bias.
Types of writing should be compared only against a comparable type of writing, controlling for the education and social class of the author, using comparable methods to a proper cross-sectional analysis. This is commonly done and then the text is placed into a software program to create a text analysis, so there this an "objective" answer to this question, but we simply do not know what it is at this time. For example, a speech by Abraham Lincoln could be compared to a speech by Barack Obama; a letter written by Sherman to a letter written by General Petraeus; a letter written by a low ranking army conscript with little education to an email written by a modern high school senior. I feel certain that the differences in style would remain quite striking. This should not be surprising, since language, art and writing styles change over time due to social influences.
No one today would say, as per the OP:
Do not hurry, men, and fire too fast—let them come up close before you fire, and then aim low, and steadily.—Gettysburg, p. 436
(For non-native American English speakers: several parts of this are off-putting. "Do not" as a command, "hurry" as a verb, "men," and "steadily" sounds very exact. Taken together, this phrase is highly unlikely to be uttered by a modern American)
Now, as Oldcat points out, Civil War humor is largely left out of modern historical accounts, which represents a selection bias. This has occurred because either the humor would not be understood by a modern audience or it is a dark humor, which is arguably no longer socially acceptable in most cases. Quoting major historical figures using dark humor has the effect of making them seem strange, twisted or callous, rather than heroic and ennobled, which is the effect in much of Civil War era writing. Nevertheless, it still often relies on classical literary device and the changes in uses of humor do show a change in writing styles overall.
There are mainly three reasons for the simplification of writing styles (but there are more as well): 1) changes in art movements 2) changes in education 3) changes in views about the audience. The third point runs into the first two points, but I see it as the most important so it will be examined.
The time period around the Civil War is heavily influenced by Romanticism.
The movement validated intense emotion as an authentic source of aesthetic experience, placing new emphasis on such emotions as apprehension, horror and terror, and awe
Writing styles from this time period thus appear much more "poetic" in nature, but it was soon replaced by Realism. Much of modern writing is probably properly called Utilitarian. It is also Postmodern and many American writers still rely heavily on Realism, even in more creative writing forms.
Classical education ended in the 1960s. One of the main reasons is that the sciences and "social sciences" became much more important. You are correct that Americans in the Civil War who were educated, which is a much smaller percent than today, would have had a much stronger background in literature.
By the end of the 18th century, in addition to the trivium and quadrivium of the Middle Ages, the definition of a classical education embraced study of literature, poetry, drama, philosophy, history, art, and languages.
They would have learned Latin and Greek and learned the classical structures for writing speeches, such as Eulogy or Apologia. The Gettysburg Address is an example of a Eulogy.
Attitudes Towards the Audience
The most important difference is the way the creators of books, speeches and historical accounts view the people that will read it. According to Dominic Strinati in "An Introduction to Theories of Popular Culture" there are three ways to divide art by its audience: folk art, high art and popular culture. In the late 19th century, art movements started to address art to be created for the "peasants" or the "volk." The early 20th century included many disputes about mass media creating low brow mass culture.
Today, more or less popular culture has been victorious. A major reason is that mass media markets expanded in the 20th century and we are continuously exposed and normalized to pop culture messages, but it is also simply a cultural shift. Writers are encouraged to keep communications simple and understandable to everyone in society, unless a book or communication is absolutely directed for a small number of experts. Political speeches are no longer flowery or use complex grammar, because it is considered more important that everyone understands and participates in the democratic process. Letters are largely not written at all and have been replaced by email, text messaging, instant messaging, or even picture messaging.
The "goodness" of a writer is now measured by how well they communicate a message (so much so de Toqueville is being called a bad writer!?). During the Civil War, writers would have been much more concerned with the style of their writing. Style can also be used to communicate messages.