Clethra is a genus of approximately 81 species of deciduous or evergreen shrub of small three, in the family, Clethraceae, in the Heather Order (Ericales). Various species of Clethra are native to eastern North America, the Caribbean, Mexico, Central America, South America, southeastern Asia, China, Tibet, Hainan, Korea, Japan, Philippines, Indonesia, and New Guinea.
There is also a distinctive evergreen species, Clethra arborea, that is endemic to Madeira. There is a standing claim that it once occurred on Tenerife in the Canary Islands, but is now extinct there.
The Clethra discussed in the OP is not identified to species. However the photo of the dry fruit does look like that of Summersweet (Clethra alnifolia), a deciduous shrub native to Nova scotia and the eastern and southern U.S. to as far west as Texas.
Clethra alnifolia is the most commonly cultivated Clethra in the U.S.
Clethra seeds are small, and numerous within each capsule. Literature says that Clethra have three locules per ovary (later the capsule) and 20 to 40 ovules per locule. So if every ovule was pollinated there could be 60 to 120 seeds per capsule, though actual pollination efficiency is rarely 100% and could be much lower. It is possible that the fruits obtained in the OP had few or no viable seeds, either due to lack of pollination, proper development, or seminal herbivory.
