using a gigabit network switch with a wifi extender

I want to be sure I understand what you are doing. I think you are trying to use this Netgear WiFi Range Extender model EX-6150 not as a WiFi extender, but in its alternate function as a WiFi Point of Access for a small wired network in your new game room. The game room network will have several computers connected to a switch, and then the model EX-6150 extender also plugged into the switch. The EX-6150 will be configured as a Point of Access device, and will communicate with another Point of Access device elsewhere that is connected to the internet via another network. In this way your game room will have internet access using a WiFi "bridge" between your internet connection point and your game room.

That may not be exactly what this Netgear EX-6150 was intended for, but I think it should do that anyway because it CAN be configured to be a Point of Access. However, when it is set that way, it cannot also be a WiFi Extender. That is, when acting as a Point of Access, its source of incoming signals that can be re-distributed to WiFi devices is the wired connection to its only ethernet port. So, although it will get info from the remote WiFi signals of the other network and feed them to the switch where other game room wired devices can get that, it cannot also serve a new group of Wi Fi devices in the game room. At least, that's what I think it will do.

To make the EX-6150 work that way, here's a link from the Netgear website on how to configure the EX-6150 to be a Point of Access device.

https://kb.netgear.com/30331/Configuring-EX6150-as-an-Access-Point

You need to understand the speed capability of these devices. We have a similar one made by Ubiquiti, their model 802.11 AC Pro POA device. It also uses dual bands (2.4 and 5 GHz) and 3x3 MU-MIMO technology. IF I understand it correctly (not guaranteed!) it can support nine simultaneous devices, each with 150 Mb/s communication speeds, for a total maximum throughput of 1.3 Gb/s. BUT that does NOT mean you get over 1 Gb/s through it from one device to another, so it is NOT a 1 Gb/s network connection channel. If we assume that the other network (the one with the internet connection) can also communicate at 150 Mb/s, then there will be ONE communication channel of that speed linking that network with your game room network, and that will be the limit on your access to the internet. Of course, the speed of your internet connection itself may also set a lower limit. For example, in our home system we have the Uniquiti unit on a wired gigabit ethernet system, but the internet access on that network is a fibre optic connection for which we are paying to get 75 Mb/s download speeds. (We could buy more, but it's not worth it for our use.) So although the many wireless devices in our home can use fast connections to other wireless and wired devices on that internal network, access to the internet outside is limited by the speed we're buying on the fibre optic system. For us, that's plenty fast enough. But we are not trying to support many game players who demand high internet connections speeds simultaneously.