A non blue eyed person will see that there are 100 blue eyed people and observe them killing themselves on the 100th day. For the blue eyes, if the 99 blue eyes they can see were all of the blue eyes, they would kill themselves on day 99. So the different eye colors have different information.

As far as I can see, the brown eyes all kill themselves the next day.

They won't, because they only know that everybody else has brown eyes. They do not know if they themselves have brown/pink/green/purple eyes. They only know they don't have blue eyes.

Because they would see 100 people with blue eyes, not 99, and so wouldn't kill themselves until one day later, at which point they now know they have brown eyes, because of the suicide.

> at which point they now know they have brown eyes

.... and so have to kill themselves

The islanders don't know that there are only blue and brown eyes. For all they know, their eye color could be green, red, etc. Hence, the logic only applies for the blue-eyed people.

It would never reach that state, because every blue eyed person would have committed suicide on the 99th day.