Alright, I'll play the devil's advocate.
Unfortunately, the records aren't complete enough to disprove my theory either. Often you will have to use naming patterns, circumstantial evidence, location correlations, and other factors to make progress. Of course, it is just assumptions. I never said it was anything more. BUT, it's a lead to a line that would otherwise be a dead end with nowhere else to look.
But, to go down the line of thought that John T. Gibson COULD NOT be Tryon, which certainly could be the case, then let's look at the situation:
1. Caroline Braton Gibson is born on 28 Dec 1817 in Tennessee, and she consistently lists her parents' birthplaces both as Virginia. Of course, this is a strike against Tryon being her father.
2. John Gibson and Margaret Givens (whose mother was a Bratton) were married in Virginia in 1813, and are excellent candidates for Caroline's parents, based upon Margaret's mother's maiden name, her birth place, and the date of marriage in relation to the date of birth.
3. A John and Margaret Gibson are buried in the same town where Caroline settled, as well as at least one other presumed brother or relative, Leroy T Gibson.
4. Leroy T Gibson gives his parents' birthplaces as NC and VA consistently. This is interesting, because though Tryon Gibson says he was born in SC, he was orphaned and raised in NC. Not a perfect match, but an interesting correlation.
5. Both Leroy and Caroline (and other presumed siblings) were born in Tennessee to parents from Virginia, they migrated to Missouri, then on to Goliad Co, TX. Tryon's family followed this exact same pattern, so it would at least warrant a look to see if he moved to Goliad County as well.
So, it's pretty certain that Caroline Braton Gibson's parents were John Gibson and Margaret Givens. It's very likely that the John and Margaret Gibson buried in Caroline's town are her parents (To assume that Caroline's parents are John and Margaret Gibson, and yet she has no relation to a John and Margaret Gibson buried in her town, when there are only a couple hundred residents there at the time seems a little silly to me). So if that's really the case (and I could be wrong), her father's name was John T. Gibson. At the very least, her father John T. Gibson followed Tryon's family's migration pattern almost exactly, and they would have been close relatives. I think it's worth a second look to see if John T. Gibson couldn't have been Tryon. It's definitely possible.
Now let's take a look at the association between Caroline's family and her husband's family. Caroline married a son of Thacker Vivion (In Thacker's household in 1850 is a 17 year old male named Tryon Vivion, possibly a son). Caroline Braton Gibson and Lloyd W Vivian were married about 1834 in Missouri. In 1834, the Thacker Vivion family is known to be living in Jasper County, Missouri.
"The Biographical Record of Jasper County, Missouri" states that Thacker Vivion settled on Center Creek in 1831, and that in 1833 there were four Gibson households that also settled on Center Creek: Tryon Gibson, Isaac Gibson, William Gibson and John W. Gibson. It's very probable that Caroline Braton Gibson is the daughter of one of these Gibson men, met the son of Thacker Vivion living nearby, and married him around 1834. If Caroline's father's name is John T. Gibson, it pretty much rules out Isaac and William, because no parts of their names match. That leaves Tryon Gibson and John W. Gibson. So, if one was to guess, just for fun, which one was likely her father, would you pick the one with the wrong middle initial, or missing the first name?
If in fact Tryon's first name was John, and there was another John Gibson living in the area, it would make perfect sense to go by a middle name in order to not confuse the issue.
Again, nothing solid or concrete, I'll freely admit, but a whole lot of circumstantial evidence that needs to be disproven before it can be discarded. If nothing else, I'd say that a search for information on John Gibson and Margaret Givens would help to disprove the Tryon Gibson connection, if you're certain that it's not him. Happy hunting!