If you look at the current listing for Alexander Hodge who served from SC,http://18.104.22.168/DAR%5FResearch/search_adb/default.cfm
You will see that the listing says he died in Columbus, Colorado Co, TX 17 Aug 1836, and by looking at the items from clicking on the DAR Member number links, the service claimed was under the Swamp Fox.
The latest applicant under him has an old DAR number, I think from the 1960s, but most importantly the listing is 'flagged' with the notation that future applicants must prove service. There is no statement of service or reference to a pension application file. The first member under him has a number that may indicate application before the 1920s.
So this is one of the many thousands of instances where DAR has discovered an error: in service attributed to a particular person by previous applicant-members, or in applicants' accounts of descent. Just as in this example, in the days before DAR adopted more rigorous evidence standards for proof of service or proof of relationship, many mistakes were made in the course of drawing conclusions from looking at a limited range of evidence.
Sidebar: I know of this from experience regarding one of my ancestors. He spun quite a tale in his pension applicaton, and the Pension Office granted him a pension. However the service record he claimed shows desertion after about 6 months in the Continentals. DAR (in 1940s-1950s) accepted the pension file as proof of his service, and never required real evidence that the applicants even descended from this man. I actually suspect that my ancestor claimed another man's service, but was not aware that the other man had deserted.
You have concluded that the person whose pension file you looked at could not be your ancestor. I suspect that at some time in the past 30 years, the DAR genealogical staff drew the same conclusion.
So either your ancestor did not have the service that was said by family members, or he actually did but another person by the same name claimed his service.
Not being a member of your family, it is not difficult for me to look at the possibility that someone in the family did some superficial genealogical research and found a name on a list (whether a militia roster, an index to service records in the Continentals, or a list of pensioners, such as the one published for Congress in the 1800s). The genealogical mistake, "same name = same person" is the most common one, and was much more common 50 and 100 years ago than it is today. DAR and other lineage societies accepted such reasoning long ago.
Naturally the local DAR chapter to which probably some members under the old Alexander Hodge applications belonged, would have accepted what National accepted. It is likely that the local chapter Regent(s) assisted with the applications in the 1940s-1960s, even back to the earliest one. With more than 20 women entering DAR under your Texas ancestor, it is no wonder that there would be a strong family tradition.
I can't tell from the listing when National Society DAR concluded there was a problem with the proof of service, but it could have taken place around the time of the Bicentennial, when many new applicants presented disproofs of descents or of service that had previously been accepted, but which were based on faulty research.
I know it may be difficult to accept that all the past local DAR activity might have been based on faulty information, but this is really not uncommon.
As another sidebar example, one of my ancestors had a local chapter place a DAR marker on his gravestone in the 1960s. Yes, his rather common *name* appears on a militia list, but no one ever (successfully) applied for DAR membership under that or other alleged service on his part. I would never be able to learn if there had been *unsuccessful* applications No one in the chapter placing the marker was even one of his descendants. How placement of the marker came about is a bit mysterious; National Society is supposed to approve such placements, but repeated queries in different directions have not resulted in any explanation for it.
If I were in your place I would contact the nearest DAR chapter and ask for assistance with investigating whether your ancestor was actually the one with the service claimed by another person. It really did not happen often, but it is not impossible. As you must realize, it would also be very difficult to prove.
By the way, "Swamp Fox's" name was Francis Marion ('Fox' part of his nickname, not his surname). You can look at a quick biography on Wikipedia.
Sometimes we simply have to accept that since we are all human, we make mistakes, and so do others. If today there were a DAR membership applicant under the ancestor mentioned in my first sidebar, above, the conscientious DAR Chapter Regent would suggest that she pick a different line to research for possible Patriot Service. Perhaps you can look at this possibility as well.
I wish you good hunting!