Yes, he is allowed to work. The person at the job centre has no idea what they are talking about. If his visa did not allow work, it would be stated specifically on the visa itself.
If the visa says: "No work or recourse to public funds", he can't work.
But if it just says: "No recourse to public funds", he can work.
No matter where the visa is issued, it allows work - it's about the type of visa (SPOUSE/CP), not the place of issue.
While pretty much all visas are sponsored - you sponsored his spousal visa, while a university or a company will sponsor a work/student visa - it's not possible for someone to have applied for him from inside the UK, as he has to apply himself from his country of residence.
Sheffield only processes settlement visas, not sponsored work or student visas... so he could not have applied for a different visa in Sheffield. If it was a sponsored work or student visa, it would have an issue place of New York (and would still allow work).
The UKVI website specifically states that he can work on the spousal visa:
What you can and can’t do with your visa
You can use your ‘family of a settled person’ visa to:
- work (you can’t work if you’ve got a visa to get married or to become civil partners)
- study
You can’t get public funds for yourself and any dependants.
(
https://www.gov.uk/join-family-in-uk/overview)