Expanding on Edlothiad's detailed answer, it should also be noted that the bearers of the Elven rings were all, at a time or another, members of the White Council, while Thranduil was not.
[...] and in that time was first made the Council of the Wise that is called the White Council, and therein were Elrond and Galadriel and Círdan, and other lords of the Eldar, and with them were Mithrandir and Curunír. And Curunír (that was Saruman the White) was chosen to be their chief, for he had most studied the devices of Sauron of old. Galadriel indeed had wished that Mithrandir should be the Lead of the Council, and Saruman begrudged them that, for his pride and desire of mastery was grown great; but Mithrandir refused the office, since he would have no ties and no allegiance, save to those who sent him, and he would abide in no place nor be subject to any summons. But Saruman now began to study the lore of the Rings of Power, their making and their history.
The Silmarillion, "Of the Rings of Power and the Third Age"
While the text reports that in this first meeting of the White Council there were indeed "other lords of the Eldar", it seems clear that the most important of them all were Elrond, Galadriel and Cirdan, who ruled the three most important realms of the elves in Middle-earth, respectively Imladris (Rivendell), Lothlorien and Mithlond (the Grey Havens).
Compared to these three locations, the Woodland Realm ruled by Thranduil was far less important, and both him and his people were rather isolationist and far less involved in the great events of the Second and Third Age of Middle-earth, having a notable part only on the Quest of Erebor and the Battle of the Five Armies (both of them described in "The Hobbit").
It is possible, and in my opinion, very likely, that Thranduil was present during this meeting as one of the "other lords of the Eldar", but his stature is not high enough to be explicitly mentioned by name; Elrond, Galadriel and Cirdan instead are explicitly named, and they are the first bearers of the Elven Rings.
Besides Elves, the other most important members of the White Council were Gandalf (Mithrandir) and Saruman (Curunir): the stature of Gandalf is further confirmed by Cirdan giving him his Ring, when he arrived in Middle-earth, so it passed from one of the most important Elves to one of the most important Istari. All of them had a large and active part in the struggle against Sauron, while Thranduil was king of a regional power at best and far less prominent in the grand strategical layout.
It should also be noted that the Rings were not mere ornaments or symbols of regal status, they were effectively instruments to be used in the war against Sauron, even if not directory in battle they had a power that their bearer could actively use to pursue their goals. This is the reason why Cirdan gave Narya to Gandalf (it could better serve the cause in the hands of the Wizard) and why the other two were in the hands of the rulers of the most important realms of the Elves, not just from a prestige point of view but for strategical reasons as well.
In this respect, the verse that literally reads "Elven-kings", should be meant as a more generic Elven lords (and it is curious that the "Dwarf-lords" who received the Seven rings were instead titled Kings of their own realms).