I was just looking through notes I took down at an event to transcribe and start sorting them. I was listening to the soundtrack to Hamilton while I was doing this, a musical about a man who wrote like a fiend.
In fact, his voraciousness as a writer is a big reason why I hold him in pretty high regard. He was a man who understood what it feels like to have an idea in your head that you can't push out into the world fast enough. He really wrote like he was running out of time, a feeling I can sometimes sympathize with.
But as I was writing this out, I realized something incredibly important: Hamilton wrote at the insane pace that he did and all before typewriters existed. He wrote every single thing he ever wrote with a QUILL. Not a pen.
#A QUILL.
Yet his handwriting is ridiculously formal! I get that people back then had to write everything without a typewriter, so they had a lot of practice. But there are two clearly identifiable states of Hamilton's penmanship, or really anyone's penmanship: when they are attempting to write as properly as possible, and when they are prioritizing time over clarity.
Even Hamilton's second state of penmanship is excellent! I would not be able to read my secondary handwriting if I didn't read it within a week of writing, otherwise all the context integral to the act of deciphering it would completely disappear.
Hamilton and writers of his time wrote with amazing clarity and style — we clearly don't.
Is it entirely because of the introduction of keyboards, or was there something else going on?
Have we only seen the best writers of the period, because the rest of the population was poor and didn't know how to write? Do we see bad handwriting today because it has become so completely spread (at least in first world countries) that everyone can do it, and the introduction of keyboards hasn't made us get any better?
Or were people back then thinking about handwriting, language, and communication in a completely different way than we do today?
TLDR
Why did Revolutionary period writers write so damn well? Or perhaps the better question is, "Why didn't they write poorly?"
Speaking as someone who works with a lot of handwritten stuff from the 18th and 19th century, I can assure you that not all of the handwriting from the period is good. Handwriting was a taught skill, and (weirdly enough) I was just at a presentation today on different styles of handwriting from a bit earlier periods (mostly 16th and 17th century). Good handwriting was considered important especially for those who would expect what they were writing to be read by many other people - those who were well educated or politically active. Others, like merchants, tended to write in whatever way was accessible to them and those with the immediate need to look at what they wrote. Their handwriting (and I speak from a lot of experience) was often very difficult to read.
The important thing to think about is not only did fewer people read in the 18th century, but many people read but could not write. This is actually a question of a lot of debate, though I won't go into it here. The thing to remember is that handwriting tended to be based on who was the audience, and level of training. Most revolutionary figures were well educated and had wider readership (or needed to write clearly enough so that printers wouldn't screw it up), so they tended to have good handwriting.
As an aside, why on earth is doctor's writing incomprehensible? Are they using some kind of insane shorthand taught in medical school?
Ball-point and felt-tip pens allow for both faster and continuous writing.
Dip and fountain pens require slower/smoother writing to get even ink distribution and you are generally restricted as to the direct of your stroke (going in certain directions can cause the nib to 'skip' or 'stutter' which can 'shotgun' little drops of ink onto the paper, it can also cause the nib to need to be realigned or even break them depending on the material).
While I can think of no texts that directly answer the question, for some years I did do illumination as a hobby in the Society for Creative Anacrhonism and experimented with traditional dip pens and fountain pens with various nibs as well as more modern felt tip pens/markers with tips designed to mimic the former. With a dip or fountain pen my strokes had to be far more deliberate and done with an even tempo to avoid the 'shotgun' splatter of ink and to keep the lines uniform.
Another thing to consider is inks in most modern ball-point pens is fast or 'instant' drying whereas even most modern inks still used in dip or fountain pens (although plenty of ball-point pens still smear/smudge).
With the first commercially economical ballpoint not showing up until the late 1930's (http://www.todayifoundout.com/index.php/2014/08/origin-ballpoint-pen/) and Biro's gravity-fed ballpoint in in 1943/1945 (http://www.freepatentsonline.com/2390636.html) we still really didn't have a ballpoint resembling what we have today. It isn't until the mid 50's that we see Bich develop what we know today as the Bic pen (patented in 1966 PDF WARNING http://www.freepatentsonline.com/2734484.pdf).