Each Google Wifi point is a router that's designed to work with your existing ISP and modem. Google Wifi can create a mesh network only with Google Wifi and OnHub routers. Google Wifi will work with most connected wireless devices including tablets, printers, and smart appliances.
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When a Google WiFi unit is set up in bridge mode, it does not support mesh/secondary units. It has to be in router + firewall (NAT) mode to support mesh/secondary. When it is set up this way, it will be creating a new "inner" network separated (by the firewall) from the "outer" network it is connected to (the one created by your existing ASUS router, in your case). Having some devices connected to the outer network and some devices connected to the inner network will result in them having difficulty communicating with each other (the auto-discovery protocols in particular won't work across a router/firewall).
Lastly, just to be clear, Google WiFi is not a WiFi "extender". The primary unit must be connected to the internet via Ethernet.