I can attest for one, as has been mentioned above, that the prone position was certainly used and perhaps briefly taught. Take a peek at this vid for an example of the process:
https://youtu.be/bcKWZJQ2AWA
It was also used on several occasions by both sides at Antietam, let alone countless other examples throughout the war, in the former case in the Miller's Cornfield and at the Sunken Lane where soldiers went prone either to avoid artillery and musket fire, or to surprise the enemy respectively. So it does have plenty of tactical use, as well as plenty of historical footing, as well as a good amount of cons to it's use.