Games, programs or workloads that it will be used for: DayZ @ 1080p, 4k video playback
Other details: I built my current PC back in 2012-ish, showing its age now and the GPU just died. Specs:
Intel i5 4670k
GTX 780 3GB
Cooler master hyperX 212x cooler
It suits my needs fine other than the dead GPU, is it worth upgrading to a more modern platform or just get a used GPU, like a GTX1080 or something?
I think you're best off replacing the whole system, especially if you're playing something like DayZ. The reason being that those sorts of games are more CPU bound than GPU, especially when it comes to average framerate.
4790k isn't far off from a 4670k and Warframe is a great single example to demonstrate this limitation. Same concept carries over to any game where enemy/player positioning is negotiated across the network.
Worst case scenario, you buy the GPU you're eyeing and if the experience still isn't acceptable, look at ~$250-$300 for a new platform as well.