For me personally, they became a lot more relevant in the later half of the PS3/360 generation because multi-platform games were running really bad on console. I remember playing Borderlands and fighting those spider ants with an SMG. The frame rate felt like it was dipping down into the teens with all the numbers popping off the enemies. I thought to myself "I paid 2K for a TV, $400 for this console, and it looks like a smear of stuttering jelly across the screen..." I think it was a week or two later when I saw a DF vid on performance that compared the consoles and PC.
The combination of Steam maturing into a nice platform, Microsoft making the Xbox controller easily supported on PC, and hardware that could easily run games at 1080/60fps really caught my attention and got me looking at the PC more and more. Once I built a dedicated gaming PC with an SSD in 2011, there was no going back.