First, on the golfer and his relationship to other Sneads from Virginia. If there is a connection to the golfer's family tree, I don't believe it would have occurred more recently than 5 generations before your John L. What I know of the golfer's family history seems very clear back to mid-1700's Hanover County, VA.
My father maintained that we were related to the golfer, but I now know from research that it would have been way back, if at all. Only a few Sneads came to Virginia and produced descendants, so that seems to be the basis for people's belief that we are all related. The records don't support the idea that there was one common Snead/Sneed/Sneyd ancestor.
Second, on tracing your family history back from John L., I can provide some observations. You can decide if they amount to useful help.
John L. was 26 in the 1860 census and in a household with William E., age 21, and Sally A., age 15 and others named Humphrey mentioned below. In the 1850 Fluvanna census, there is a household (dwelling unit 358, family visited 359) comprised of Lucy, age 34, John L.C., age 11, and Sallie A., age 5. The questions are whether these are the same Johns and Sallies and where did William E. come from.
John's change in age from (apparently) 11 to 26 from 1850 to 1860 raises a question that may be explained by inaccuracy in the initial reporting, the initial recording or the interpretation of the old records currently. I have seen this sort of age inconsistency on other occasions when it is clearly the same person from census to census. Sally's presence in the household with a consistent age change supports a conclusion that this is the same John. William E.'s absence in 1850, and appearance in 1860, is unexplained.
The 1850 household certainly looks like a one-parent family.
The Albemarle County, VA (this county borders Fluvanna on the west) marriage records show the marriage of a William Snead and a Lucy M. Humphrey on 12/14/1834 (probably the date of the county license) or 04/22/1835 (probably the date of the ceremony). If this is the Lucy shown in the 1850 household, then she was married early enough to have been your John L.'s mother, whether he was age 11 or age 16 in 1850.
I don't have any information to account for the absence of Lucy's husband William from the 1850 household, but death seems the likely explanation to me.
Noting that Lucy was a Humphrey, I think some more clues are to be found in the 1860 Fluvanna census. Note that John L. is then in the household of Meredith (age 75) and Celia Humphrey (age 78). Seems likely to me that these are Lucy's parents. How to explain Lucy's absence? Again, probably death.
It is interesting that the Humphrey household in 1860 was a neighbor, four doors away, from a household comprised of John H. Snead, age 64, Susan, age 48, William R., age 16, and Sally M., age 14. A John H. Sneed and a Susan Robinson were married in Albemarle County on 11/12/1839.
The two Albemarle marriages, the proximity of the families in 1850 (Lucy's household was 9 families away from John H. and Susan's) and 1860, and the commonality of the children's names suggest to me a close relationship between between your John L. and John H., either nephew-uncle or grandson-grandfather.
For John L.'s father, I would look for a William from Albemarle, perhaps with a brother John H. I have studied Fluvanna and Albemarle Sneads closely for my own search, and I do not recall seeing this combination in an unassigned or available family tree, but there are several persons for whom I do not have a descendants' list, so the possibility remains. William and John were very popular names in Snead families, making it hard to distinguish one from another.