Ok, real, final solution!

Going to Device Manager > Network Adapters > Intel Dual Band > (Right Click) Properties > Advanced Tab, and setting "802.11a/b/g Wireless Mode" to 2.4Ghz 802.11b/g solves the problem!

Looks like the problem was on the 5Ghz WiFi network and that is why changing the WiFi adapter helped and probably why I never had problem outside of this PC (since my other computer always use the 2.4Ghz band anyway).

The remaining question now is why FFXIV took problem with my 5Ghz WiFi, it is true that Google Wifi has a exotic standard that merge the two bands in the one Wifi Network and that might have helped but I still have no idea why should this matter to FFXIV.