The distinction they're drawing isn't WLAN vs LAN but WAN vs LAN. Remember that in the peak days of the LAN party it was an alternative to gaming on dial-up or DSL - even if you had a good connection it was unlikely the whole game had one. A reasonable Wi-Fi connection today is miles beyond the WAN connections of Y2K.

The big LAN parties also had a fat internet connection at a time when the best you could have at home was INSD. But even more important was that everyone at the LAN shared their piracy collections over DC++ or FTP. Good times.