In network terminology, Router 1 expects to be the only router in the house without advanced configuration.
Networking involves 7 layers of protocols, routers operate at Layer 3 (L3) and use TCP\IP Addresses (192,168.1.x) to send and receive data. But, in bridge mode, the Layer 3 routing function is disabled, and all routing occurs on L2 (Layer 2) using a MAC address such as 08:6d:41:e7:cb:20. Layer 2 can only be routed to hosts that are in the same local subnet.
The problem with not seeing your air* devices has to do with layer 2 routing\discovery. Air* devices basically send a message out to the network advertising that the service is available on a given host address, every host on the network sees the broadcast. Some routers disable this sort of broadcast as it can become quite chatty on larger networks (probably the case in your situation).
The router 2 in your setup, if acting as a bridge, will retransmit broadcasts to anything connected to it, including the WAN port to the main router. But, if the main router doesn't retransmit, it dies there... but anything connected directly (ethernet or wifi) to Router 2 will see the retransmissions.
If your Router 2 is setup in default mode, you run the risk of NAT, DHCP conflicts and might see intermittent outages, not just on your end, but the entire house. It looks like that model has a bridge mode Connection type in the internet setup, so instead of using the default DHCP Auto Config connection type, if you set it to bridge mode, that should do the trick. It will then act like a hub\switch for anything that connects directly to it, either WiFi or hardwire.
In short, a router operates at L3, but a bridge\hub\switch operates at L2. If all your devices were ethernet capable, a switch would do the trick, but if you also need WiFi, making the router a bridge L2 device accomplishes the same thing as a switch would for wired devices.