That's just not true. All but a handful of the smallest LAN parties I attended I the 90's and early 00's included console gaming. It was pretty much assumed there would be consoles at a LAN.

Halo, which was arguably the first professionally competitive game, had it's entire birth at LAN parties as it did not support online play at all and had to be done with a LAN.

Smash brothers had a similar Genesis story at LAN parties I attended. LAN continues to be the preferred competitive environment for every game due to latency.

I'll tell you what killed LAN parties I attend, it was the year steam became standard. To get everyone at the LAN to play the same game no longer meant passing the CD around, but in /socially pressuring your peers to make a digital purchase/ That was the death of LAN.