Thank you for providing additional details. What a quest to take on, but I would imagine it is very rewarding and interesting.
Have you checked to see if the soldiers possibly knew each other before signing up? Did any live near others during the 1860 census? If men chose to sign up with Company A, two or more friends could have encouraged each other to sign up together. Have you found any other soldiers born in or near Edgefield?
Your research seems extensive so you’ve most likely run across the following, but suggesting it in case it helps:
https://www.cem.va.gov/nationwide-gravesite-locator/I believe some of the headstones include company or battalion numbers.
You’ve most likely thought of and run across this, but the ages provided by the men could be inaccurate, allowing them to serve if they were too young or too old.
Have you found any men who had shady backgrounds? Were they allowed to sign up? Is it possible that Peyton signed up with them?
You’ve most likely searched for various Smith spellings for Peyton? (I tried, but obviously it didn’t help.) Have you searched for Smiths in the Edgefield or South Carolina area to see if boys or girls had “similar” given names? Perhaps Peyton was a surname in the family and the parents gave Peyton’s siblings similar names, such as Gordon Smith or McKenzie Smith? Or, as previously suggested, he switched his name around. Possibly his surname was Peyton and wanting to cover up his past he changed his surname to Smith.
Apologies for all of the questions; I’m hoping that I’ve asked one or more points that you’ve not yet considered, or perhaps they may make you think of other ways to search. Other RootsChatters may come along with better ideas.
Added: perhaps it’s been mentioned but did anyone else in his company desert around the same time or place as Peyton? Do you know if Cottonwood Crossing was a notorious place for desertion and if so, have you researched where other deserters ended up? Of course, they could have scattered in all directions.
I will have a look around for details for G. W. Smith and family.