Hello NZvixx,
You have the correct birth cert with dob 11 March 1874 for your gt. grandad - the son of John E. Rogers & Nellie F. Leown. As you have explained the birth cert does not have the name of the child.
I researched the records of the US Consulate, Manchester, England which are held at the US Archives, Washington, DC. I found more records for your gt grandad. Besides the US WWI draft registation for James T. Rogers there are documents for
Application for Registration - Native Citizen - 2 pages
Affidavit to Explain Protracted Foreign Residence - 2 pages
transcript copy of Return of Birth, Dept of Health, City of NY
I can email you these documents if you want to send me an email address.
Your mystery remains - where is this family of John Rogers, wife Nellie & son James in the 1880 census & any other NYC records.