Why do men's suits have buttons on the sleeve?

During the Napoleonic wars Gen. Napoleon demanded that his troops be clean in their appearance. They had this nasty habit of wiping their noses on their sleeve, so he had buttons sewn on the inside of the cuffs to prevent them from wiping either their nose or their mouth clean.

http://www.4masonry.com/education/files/artjun02/ANCIENT%20SYMBOLIC%20PENALTIES.htm
gives another explanation.

The buttons on the sleeve of a man's jacket and the little slit under the buttons are symbols of the time a man's jacket unbuttoned all the way to the shoulder so that he might have easy use of his sword. The vent at the back of a man's jacket is a symbol of the time soldiers rode horseback. The vent allowed their jackets to fall on either side of the riders' legs and so keep his powder dry to more effectively kill his adversary.

According to Readers Digest, the fashion at the end of the 18th century was for cuffs so narrow and long that they needed a slit in them so they could be unfastened and fastened to let the hand through.

At one time, supposedly, coat sleeve buttons had an eminently practical function. It seems that Frederick the Great, ruler of Prussia from 1740 to 1786, used to enjoy nothing more than the sight of his troops neatly decked out in uniforms and lined up in rows. Only one thing spoiled the scene: the soldiers insisted on sweating, getting dirty, catching diseases, and bleeding profusely.

Since no one had the foresight to provide the troops with Kleenex with which to mop their brows, the soldiers made do as best they could with their coat sleeves. After a hard day's skirmishing, said sleeves would be covered with unsightly blots and blemishes, and perhaps a vital organ or two.

Naturally this was unacceptable. Frederick pondered long and hard on what to do. Finally, the solution (or "der zolution," as he more likely put it) dawned: sew buttons on the top sides of the sleeves, and soldiers would scratch their faces open every time they tried to use their coats for a handkerchief.

https://www.straightdope.com/21341270/why-do-men-s-suits-have-buttons-on-the-sleeves