FDR's WPA was designed to provide a short-term bridge of employment to help the millions of people who became unemployed due to the reckless economics of the previous decade leading up to the Great Depression. It was not designed to provide permanent careers. Once the economy was able to provide sufficent employment for those ravaged by the economic policies leading to the Depression, the WPA was ended. There is no reason to form a union for temporary workers who were grateful for the temporary work and had no intention of continuing that work as a career.
I'd say FDR played it just right and accomplished the predetermined goal of temporary employment for the millions of people who needed it. Clearly FDR was a reasonable man who believed in fairness. Fairness for unemployed workers during bad economic times and fairness for businesses who would need to one day hire them again when times got better. I suppose FDR was not the radical leftist that he is so often accused of by misguided conservatives, but he was sure no economic royalist either (in spite of his privileged upbringing). Go figure.