But if you have True Seeing on, are you aware that the wizard is invisible? Do you still see illusions, but are aware that they are not illusions?
If True Seeing really functioned like you interpret, Thanee, you should be totally unaware that the wizard in the corner is invisible, or that an illusion exists.
Think about the following scenario: You have True Seeing on. You turn a corner and ten feet in front of you is an illusury wall. If True Seeing simply negated the illusion, you would walk through it, totally unaware of its existence. If, however, you percieve the illusion, but are aware that it is an illusion, then True Seeing has functioned NOT as an illusion-negator, but as an information-gathering spell. You can now impart this information to your companions.
The same with invisibility. If you had True Seeing going, and an invisible rogue was approaching, and if all True Seeing did was negate the invisibility, it might not even occur to you to warn your companions. After all, you are unaware the rogue has invisibility on, he's right there, you can see him. It might take you a while to realize your companions can't. If, however, you are aware that the rogue has an invisibility spell on, you would know to warn your companions, and you could do things like pretend you don't see him to lure him closer. It will have become an information-gathering spell, not a form of counter-magic.