I worked at a place that had a cube farm with white noise generators to suppress background noise. These things triggered the small amount of tinnitus I have, also some co-workers complained. The frequencies they put out definitely matters- ones with no high pitch output are definitely better.
(pretty sure my Tinnitus was from a certain Metallica concert..)
It’s the same with ANC. It’s still noise pumped into your ear. So even though you may perceive it as silence because it cancels out the sound waves entering your ears, it’s still sound entering, and thus facilitating tinnitus when exposed to it too long. Even though you feel it’s been 8 hours of silence.
The way you wrote that, doesn't make sense. The idea behind ANC is that you have some signal, sin(wt+p) from ambient noise, and to it you add -sin(wt+p). The sum of these is zero. You perceive it as silence because it is: you have added the exact opposite of some signal to cancel it out, and the result is zero, no signal whatsoever.
It would be easy to test and demonstrate that the ANC system is properly band-limited and doesn't cause any short term (perhaps imperceptible) noise spikes. I would think any engineer designing this kind of gear would be wary of causing hearing damage by making a design that works the way you suggest.
(pretty sure my Tinnitus was from a certain Metallica concert..)