Actually, the font commonly seen on typewriters wasn't Courier. Courier was a monospaced font designed to emulate a typewriter font, but the typeface used on most typewriters usually had no name at all, and was designated by size: Pica for 10 characters per inch, Elite for 12 CPI. Each typewriter company used slightly different versions of Elite and Pica, and often those versions would change slightly from year to year.
Typewriters also came in many different font styles and font sizes, and a customer could order one in any of hundreds of different styles from Script to Techno to pinprick fonts designed to be used on checks and other secure documents. My collection includes some machines with very interesting and unusual fonts, and even one with a font designed for writing mathematical formulas.