19th Century Mormon leadership was notoriously bearded, EXCEPT for their founder Joseph Smith. Why didn't Smith wear a beard like so many other early Mormon leaders?

This is a delightfully obscure question. Smith did choose to be clean shaven in his portraits, including an oil painting done in the 1830s or 1840s and a profile by a Latter-Day Saint artist, Sutcliffe Maudsley, done in Nauvoo in 1842. However, Joseph Smith, Jr. does actually seem to have grown out his beard from time to time. The popular image of Smith as particularly fresh faced and clean shaven likely had more to do with paintings done after his death, than with likenesses made in his lifetime.

Smith biographer Richard Bushman writes that when Emma Hale, who became Smith’s first wife, met him in January of 1827, he had a small beard. In May of 1844, a little over a month before he was murdered, Smith met with two travelers passing through Nauvoo, Josiah Quincy, Jr. and Charles Francis Adams. In a reflection about the encounter published in 1882, Quincy recalled Smith having “a beard of some three days' growth.” A beard isn’t visible on a death mask made shortly after Smith was killed, so presumably he shaved at some point after his encounter with Quincy. Smith may not have had elaborate facial hair, but this suggests that he could go a noticeable amount of time without shaving.

The choice of later Latter-Day Saint leaders to have elaborate beards, while Smith posed for portraits clean shaven, was probably a result of changing fashions. In the 1850s, there was what some writers have termed a “beard craze.” The first bearded presidential candidate, John C. Frémont, ran in 1856, and Lincoln grew out his beard in 1860. The Civil War greatly increased the popularity of elaborate facial hair on men.

Latter-Day Saint leaders seem to have grown beards when they become popular for other men. While Brigham Young was beardless in an 1853 photo, he has one in another photo taken in 1855. Wilford Woodruff, Sr., who was the fourth president of the Church Jesus Christ of Latter-day Saints, had a beard as president of the church, but did not sport one as a young man in an 1849 photo. Joseph Smith simply died before being depicted bearded became seen as fashionable.