If you have forgoten, the Temple of Time is destroyed in TP. I think that this leaves the possibility that there was access to the Sacred Realm where the Triforce may have been touched and thus split.
Yes he is chosen. If Zelda or Link were to touch the Triforce, the ToP would go to Ganondorf because out of all the inhabitants of Hyrule, Ganondorf is the most hungry for power.
Someone said this earlier in the thread, that the door of time never opened on the CT.
If this is true, then what is Link doing with the Master Sword in MM?