At this point in time, I'm not quite sure that many people actually give weight to any of the Major League Baseball awards. While I cannot speak for everyone, it seems to me that most have one of two reactions to awards - they use it to cement their own beliefs, or they use it to show that the voters are incompetent or uninformed (or some other less friendly term). To me, the awards aren't entirely without merit, as they provide a topic for discussion in the period between the World Series and the real kick-off of the Hot Stove season ... and they allow me to pick apart those who are much more influential than I.
The following is how the defense of Teixeira, Cano, and Jeter stands in comparison to their American League peers. I used 700 defensive innings at the position as a cut-off as a means to include players that played approximately half of the year at the given position.
Teixeira
By DRS (+/-): -1, 9th in AL
By UZR: -2.9, 8th in AL
Cano
By DRS (+/-): +7, 5th in AL
By UZR: -0.6, 5th in AL
Jeter
By DRS (+/-): -12, 13th in AL
By UZR: -0.6, -4.7, 10th in AL
Here are the best players at each position by DRS:
1B - Daric Barton, +20
2B - Orlando Hudson, +17
SS - Alexei Ramirez, +16
And by UZR:
1B - Daric Barton, 12.1
2B - Mark Ellis, 9.9
SS - Alexei Ramirez, 10.8
By these measures, I'm not terribly certain that a case can be made for any of the Yankees Gold Glove winners. While defensive metrics are not entirely reliable, I believe that they've made steady progress in recent years - at least so much so that they can be taken with more than a grain of salt, and even more so when there is such a noticeable strand of agreement between different systems.