Virginia began keeping birth records on a county level in 1853 and continued until 1896 These were COUNTY records.......not all births were recorded for either black folk or white folk. All of the children in the household in 1880, are born in the correct time frame for their births to have been recorded, if in fact they were. The LVA has microfilm of the county records, have you looked to see if any of their births were recorded? The microfilm is available on interlibrary loan. According to the LVA website Rappahannock is missing the birth records for 1863, 1870 and 1878, but they have the rest.

Because VA did not begin keeping records until 1853...there would not be a birth record for William, unless he was enslaved and the owner kept his own personal records and then you would have had to identify the owner.

Selma

(After 1896 there is a 16 year gap and then VA begins again on a STATE Level, these are the ones you get from the Vital Statistics office in Richmond).