Considering your latest info, you may be right to go back to square one. We were looking for the origins of the elder David Hays. His 1840 and earlier documentation remains unknown, at least to me.
In reality however, we are seeking the parents of Rebecca B. Hays, and while this David seems to be a good, probable candidate, I’ve not seen any documented connection. Perhaps you have found that evidence.
There is documentation of Mrs. Puckett from her marriage in 1842 to the 1900 census. The demographic data is consistent and there is no reason not to accept that information. If she is born in Highland co. in 1824 and married in Clinton co. in 1842, then she certainly seems to be native to the local area and would probably be found there in both 1830 and 1840 census listings. And if not with David, then where? Unfortunately, I have no current sources to access 1830 and 1840 census demographics, so there may be multiple Hays households with young daughters of the correct age. However, when evidence is so scare, you need to examine anything you can find.
In looking at Highland co. listings, there is Isabella Hays in White Oak twp. in 1830. John Hays married Isabella Murphy in Highland co. in 1814. Although completely speculative, it does offer the possibility of a daughter born in Highland co. to someone other than George Hays, with his daughter, Rebecca Ann. And if by 1840, both her parents (John and Isabella) had died, she could be virtually impossible to find (living with Murphy relatives??)