I have doubts that this flagw as taken at 1st Bull Run by the 3rd Tennessee Infantry. Neither their commander nor Elzey filed OR reports for the battle but in the same volume of the OR you will find an OR report from Col. John C. Vaughn of the 3rd Tennessee date mid-June, 1861 listing a flag taken at New Creek, VA on the raid on the Baltimore & Ohio Railroad. Elzey's Brigade faced Vermont and Maine troops on McDowell's right flank.
Heritage's provenance does not prove that the flag was taken at 1st Bull Run at all although the captor Thomas Cawood was definitely in the 3rd Tennessee. I lean towards the New Creek capture personally.
The flag sold for $24,000 plus buyer's premium by the way,.
Tennessee State Museum