Jeff Alexander's WorldConnect database at Rootsweb.com has Henrietta Briggs, dau of Oliver & Hannah (Potter) Briggs, married to Obed Gwinn. He gives William & Sarah (Roberts) Briggs as Oliver's parents.
In the 1820 census for Hamburg, Niagara (to become Erie) Co, NY, we find Obed Gwinn listed beside a William Briggs,aged 26 to 45, suggesting a relationship.
Since your William had children named Oliver and William, this further implies a connection.
I don't find William Briggs in Hamburg in later censuses.Do you know where he went?