I've thought this over alot, and I've watched the last 3 minutes over 50 times, and this is what it seems like since there is def not a clear understanding of what happened. There are a few options...
I don't think Aang was truly in the avatar state, I think he finally learned how to enter it but I think when he went up he wasn't in full control sorta like when u start ur car in the cold, u gotta let it heat up, and the window defrost...
That being said, I think he was killed because everything on the show points to it, first of all the reference to dying in the avatar state, second- lightning.... third - the way the big avatar fell over, fourth - the music, fifth - Katara holding him like jesus, sixth - he wasn't breathing from when he got zapped till Katara revived him, seventh - Katara crying when her water did nothing for a second, those all point to him dying for at least a few minutes, you can die for a few minutes and not be truly dead, I think that's what happened to aang.
Now, why the cycle wouldn't be broken- Aang wasn't in the full Avatar state is one option, the glow on his forehead from the water pretty much confirms the avatar spirit is still there, I guarantee it's not just his spirit cause??the moon spirit??didn't glow like that and have the glow noise of the avatar state...??I think it was more of a wakeup call for aang to master the avatar state itself and not just master the ability to enter it and float for 10 seconds basically saying "shoot me"..
Another thing about the previous Avatar states he was in, and the avatar state the past avatars entered, Aang's previous states had him totally just super powerful with him glowing, because the avatars were controlling him. The past avatars had only their eyes glow for only 1 second- the air avatar's forehead didn't even glow when it demonstrated her... there was no huge light... that's why I think what aang did was the initiation to be able to enter it, it wasn't the real thing so technically he couldn't die in the avatar state cause he hadn't fully entered it.
I don't think the cycle is broken, that would honestly make the show worse I think because he would have to win with smarts, and non avatar stuff, besides it IS called AVATAR the last airbender... I don't think he wouldn't be the avatar to end the last third of the show...