Every single wireless display option I have ever tested has been unusable for gaming.
This applies to built-in Wireless Display on Smart TVs, Chromecast, and also (my favorite option), Microsoft's Wireless Display Adapter...
Most of them use the Miracast standard and are able to receive mirroring image with varying degrees of feature limitations, but ALL of them have perceivable LAG in common...
Built-In solutions in TVs, have only been useful in my experience for sharing pictures with family and friends, and playing music.
Chromecast are a bit more usefull and allow for good streaming videos from the source, instead of using your phone/laptop as a longer route, however, it struggles with local video playback.
Now, the Microsoft Wi-Di Adapter is actually great for media content, and adequate for extending your laptop's desktop... it works great for movies and the lag is only there for a second, once or twice in 2 hours.
But gaming is only satisfactory when using a wired solution and sadly the Xperia XZ doesn't has a wired video out option.
Old Sony Xperia devices were capable of video out through microHDMI or MHL adapters, but they were recently dropped.
I actually keep my old Xperia Acro S for my 7 y/o son as a gaming device, and he enjoys playing with several Emulators on TV with its built-in microHDMI port, and using a Wii Controller as a gamepad.
If you want to wirelessly play your games you will have to wait for the upcoming ASUS ROG Phone which has both USB-C Video Out feature and a Wireless Display adapter that promises no perceivable lag for gaming on TV.
As a tip, if you want to use any TV for gaming, even on a wired connection, you should select the TV's gaming mode under image adjustments, to avoid lag as well...
