Because writing with my right hand is even more painful. Trust me, if writing with my right hand was a pleasant experience, I'd do it in a heartbeat. This isn't just some "I identify as a lefty; therefore, I do things left-handed" kind of thing. In fact, most other activities I do, I do right-handed.

Swing a bat/club? Right-handed. Throw a ball? Right-handed. Use a computer mouse? Right-handed. Shoot a rifle? Right-handed. Write a note? Left-handed. Paint a picture? Left-handed.