The only proven Atherton line with a son William listed below. Don't know when or where this William Atherton below was born.
Are you working with a DAR chapter in your area to help you?
Good luck in your search

ATHERTON, MATTHEW
Bn. MASSACHUSETTS
Rank: PRIVATE
born. 10/1/1760 BOLTON MASSACHUSETTS
died ca 1843 ROYALTON VERMONT
service; CAPT. WRIGHT COL. DRYRY
spouses
1. ELIZABETH ADAMS
2. JEMIMA PELTON
child:
WILLIAM married LAVINA FLINT
[he may have married more than once]
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Looks like no one joined DAR using your Joel Atherton from NY as their ancestor.

The other Joel is listed below:
Joel Atherton
bn. 1764 MA
died 1844 ME
spouse Nancy Crombie
children
Crombie
Rebecca