Because of the stresses to trout caused by spawning, many don't survive the winter. Especially in infertile streams they loose a lot of body fat due to the spawn and never gain it back, causing starvation during winter.
I've ever seen much in the way of unusually high numbers of dead fish after the spawn, I think in most streams dead fish go un-noticed. I see dead fish now and then, but nothing close to being a 'fish kill.'
It is good that we have had some pretty cold weather right after the spawn though, it keeps people off streams if only for a few days. I typically curtail my fishing in the fall around the hunting seasons, because I mosly fish on public land, but also because the hatches are pretty much finished for the year on most of the streams I fish.