R. Ovadiah Yosef (Hazon Ovadiah Shabbat vol. 4, 414ff.) rules on this exact type of urn that it is permissible to use.
In presenting the question, he discusses various issues at play that are debatable:
- The water in the tube isn't boiling and therefore when water is released into the main vessel it will be cooked. However ROY dismisses this issue since it is unintentional and unnecessary for the user's purpose (דבר שאינו מתכוין, פסיק רישיה דלא ניחא ליה). Furthermore, the water in the tube isn't cold but rather it's warm, at least, and one can rely on the many authorities who hold that lach (something with fluid) cannot be re-cooked (אין בישול אחר בישול בלח). R. Yitzchok Weiss (Minhat Yitzhak vol. 10 §28) is more stringent since he maintained that although many permit water to be re-cooked once it reaches the point of yad soledet (hand recoils from the heat), there is still the opinion of Eglei Tal that water must first reach the bubbling point to be considered cooked. Likewise, he suggests that a person is concerned with the water that is released into the vessel (ניחא ליה).
- Pulling the spigot and causing some water from the tube to be released into the vessel is only an act of gerama (causation), which is rabbinically prohibited, and therefore permitted here, vis-a-vis #1, where we are dealing with a rabbinic-level issue altogether. (R. Weiss, ibid., neglects to deal with this factor.)
R. Yosef also cites R. SZ Auerbach (Minhat Shlomo vol. 2 §34:23) who, in agreement with the first two issues and R. Yosef's position, writes that after investigation found the water in the tube to be almost the same temperature as the water in the vessel and therefore does not present any problem of being re-cooked.
R. Yosef further cites R. Binyamin Zilber (author of Az Nidberu) who permitted usage of this urn, as well as R. Abba Shaul (ref. by @Shoel U'Meishiv).