The fact is that they won’t let you join the itinerary in NY, when the ticket originated in Houston.
You totally missed the point. Nobody ever mentioned joining a flight in NY if it originated in Houston.
In the case you cited, the person would fly from NY to Houston. Take the flight that originates from Houston in Houston.
THEN, on the way back, instead of flying to Houston back to NY, just get off the plane in NY and skip Houston.