First, the main book you want to consult on this, though it does not discuss the ship you mention, is Koot's Empire at the Periphery. It is recent, well written, and goes a long way in explaining the relationship between the British Colonies and the Dutch. Rupert's Creolization and Contraband is also worth a look.

There are two main reasons that a Dutch vessel would be spotted off the coast of Florida in 1706. The first is the way the ocean currents flow. This map is a rough sketch, but it contains the basic premise. Ships sailing from northern Europe first sailed south along the line of the Canaries and the Azores off North Africa. From there, they sailed west until they encountered the eastern-most islands of the Caribbean. For the British, that would be Barbados, for the Spanish it could be the mainland or Hispaniola. In order to sail back to Europe, the ships would follow the current Northwards past the Bahamas and Florida, departing east just north of Florida. British ships would often stop along the North American coast, and cut back east after stopping in ports in the Chesapeake, Delaware Valley or New England (Steele, The English Atlantic; Koot, Empire at the Periphery).

You are correct that the major Dutch owned ports are at Aruba, Bonaire, and Curacao. However, the Dutch had an extensive trading network with the English throughout the seventeenth century and into the eighteenth century. The two major Anglo-Dutch Wars in the seventeenth century were wars waged by the center of the English Empire, not by the colonies. The colonists were active participants in trade with other empires, and the Dutch were among their favorites. It was the constant violations of the Navigation Acts prohibiting trade outside the Empire that prompted warfare with the Protestant Dutch Republic. Koot describes the situation quite well - "interimperial trade was as persistent in the first decade of the eighteenth century as it had been before despite significant advances in British enforcement of trade laws and in the ability of British merchants to meet colonists' needs," (p. 181-182) Koot goes on to describe how the Dutch continued to be inter-colonial and inter-imperial traders within the larger Atlantic, especially the British Atlantic. Rupert discusses some of this data in the later period, but see her info in Chapter 4, where she describes a robust trade between Curacao and the Northern British Colonies. There is a near constant stream of ships between the two colonial hubs that would follow the oceanic currents I describe above.

So, to conclude, the reasons a Dutch Sloop would be sighted off the coast of Florida in 1706 were because that was the standard sailing route north and east from the Caribbean, and the second would be because there was a thriving trade between the Dutch and British colonies.

EDIT: Just a note, I want to clarify the reading of the line you quoted "a Dutch sloop separates the 72-gun Brillante". The Brillante was a Spanish war ship assisting the French in their assault on Charleston (Grady, Anglo-Spanish Rivalry in Colonial South-East America, 1650–1725, pp. 126-127). When the author wrote that the Dutch sloop "separates," he is saying that the Dutch vessel sailed between the Spanish and French ships, not describing a Dutch vessel with 72 guns. If this was already apparent, I apologize.