My answer is similar to @SamLey, in that we agree the cause is the higher density of sugar solution water. But I have a slightly different take. TL;DR Convection is reduced in a denser solution, so the cold water surrounding the ice isn't carried away as quickly and the ice melts more slowly.
All the data i've found shows that sucrose solutions have a lower specific heat than pure water so one would expect it to cool more quickly. (But who knows how the actual tea components affect it.)
I think the effect you're seeing is due to the density of the sucrose solution.
Ignoring the tea components, a sucrose solution is more dense than a pure water solution. As the ice cube melts in a pure water solution, the cooling water surrounding the cube sinks to the bottom of the glass and the warmer water at the bottom rises, therefore there is increased heat transfer as the warmer water breaks the bonds in the ice more quickly.
But in the sugar solution, the denser water stays at the bottom of the glass and doesn't rise at all. Without any convection currents to carry the cold water away from the ice cube, it melts much more slowly. Also note this effect is compounded by the fact that the melting ice is making the solution less concentrated near the top, so it is even less dense around the ice as it melts.
Since the denser, warmer water stays at the bottom of the glass, naturally it seems to you that the tea is cooling more slowly. This agrees with your observation that if you move the straw to the middle of the ice, the temperature gradient feels a lot greater than in the unsweetened tea.
An interesting experiment would be to use a thermometer to measure the temperature after a fixed time of the unsweetened and sweetened tea with ice without any stirring, and then perform the experiment a second time stirring every 30s or so. I would guess that the constantly agitated sweet tea will cool faster than the unsweetened tea, but the undisturbed sweet tea will cool more slowly than the unsweetened tea.