Or was absolutely all traffic to and from the Spanish territories in the New World by way of Seville?
All traffic had to go by way of Seville. In that era, all Spanish activities in exploration and colonization was controlled by the Casa de Contratación, which was based in Seville. The port of Seville was where all captains are licensed, so in theory nobody from Spain could sail anywhere without license from the Casa de Contratación.
Similarly, Portugal had the Casa da Índia with the same functions.