cool , how ?
after checking and playing around
the ps4 is searching a game in its local network but down not hop out to the second "ring" , unless u tell him to (u cant tell him to search a game via an exact ip) ,if it was possible to give an exact ip server to connect to it was possible to do that.
if the ps4 does not receive an local ip of the vpn/hamachi it wont be able to find any local games on the lan network.
it will receive internet yea, it will go out to the internet via the remote vpn server. but if it wont receive a local ip of the vpn server it wont work.
unless u give the game an exact ip to login to (such minecraft. to connect a certain server)
for example,
ur isp ip is 55.55.55.55 (to the outside world)
ur local ip (LAN) is 10.0.0.1 (connected via wifi or ethernet cable to ur router/modem/switch)
when u setup a game for example COD ghost it will search inside the local IP (LAN) so everything inside 10.0.0.1/24 (or 10.0.0.1-10.0.0.255) will be seen.
since its not possible to connect directly to a vpn with a ps4 (no app or no builtin setting) u will need to share a connection via a pc or ur phone via hotspot
now when u connect ur self to hamachi/vpn/hotspot etc.. u will get a new local ip. for example of openvpn or hamachi ip the "local ip"(LAN) that u get from openvpn will be 10.8.0.1 now rememeber u need to share that ip with the ps4. thats when it get tricky.
since the ps4 cannot connect to the VPN directly and the connection must be shared via pc/hotspot the ps4 will get the local ip of the shared host and not the vpn (hotspot ip via android, 192.168.43.1/24) so even tho the ps4 can surt via the remote server/proxy/vpn it wont be able to find games since it does not belong to the same LAN (local ips) , and the ps4 wont hop to the other LAN(local ip) .
this is my conclusion. unless the ps4 cannot be connected within the same local ips / LAN of the VPN/hamachi/xlink kai/game rooms or w.e u decide to call it, it wont be able to find the open game to join.
perhaps i am wrong. i would like to be wrong a learn a new thing.
so if its possible and know the way please share .