Having two routers, and thus two networks, will complicate things. Why not use a switch? An unmanaged switch works best with Sonos. Like this one: https://www.tp-link.com/nl/business-networking/unmanaged-switch/tl-sg105/

No.  Being upstream from your router , your modem most likely is on another subnet, and if it is a true modem, it doesn’t have router capabilities.  Either way it will put your Sonos devices on a split network.  Get a cheap unmanaged Ethernet switch.  You can get easily get a brand name (TP-Link, Netgear) 8 port switch for under $20.

@106rallye types faster than me. 

Well I don’t have two routers, what I have is a router run via one of the ethernet ports of the modem (in router mode), which itself has several ethernet ports.  Would that not negate the need to get a switch if I just plug one into a port on the modem?

Again, since your modem is upstream from your router, it is most certainly on a different subnet.  It is not possible to use the Ethernet ports on a modem which is upstream from the router to connect to a network managed by the router which is downstream.  A router controls the network by “routing” traffic to devices which are on its own network.  Since the router has no knowledge of the devices upstream, it cannot “route” anything to those devices.  It can only manage things which are connected downstream, via its own Ethernet ports or WiFi. 

Thanks.  A switch is what I will get.

Also, I presume I only need to wire just one of a stereo pair?  Does it matter which, of each pair?

The Left is the primary, if you can only wire one, I would choose the Left.

Do you know if there really is much benefit to wiring both?  I’d be looking at running a lot of wires if I did….

Not personally.  But I’ve never had the problems you are experiencing.  

Likely, in a quiet wireless environment (minimum interference) there is very little advantage in wiring both left and right. That said, I’m anal about wire and while I’m suited up for the wiring chore, I’d wire both. If, however a player is in a nearly impossible to wire spot and there is a nearby wired unit, there is not much to worry about, but each situation is unique.

In may own situation I live in an urban area, next door to a medical facility. Interference can be an issue.  One day I stopped counting at about 70 access points while scanning. Wiring is a good idea. I have a wireless pair of players separated by about seven feet, the left speaker is about three feet from an access point, the right about seven feet from the access point. Sometimes the right speaker struggles with connectivity. I’m somewhat of an experimentalist and usually keep the pair wireless, but I can wire the right speaker in a minute if I want.

As a side note, wireless capability has continually improved over the years. CONNECT and its partners are now two generations behind. Back in CONNECT’s day a home would have only a few WiFi devices and wireless CONNECT’s would be happy. In some respects they would have been the most advanced WiFi tech in the house. Now, when you start counting, there can easily be 20-30 WiFi devices (Phones, pads, computers, TV’s, streaming boxes, thermostats, wireless cameras, etc., plus SONOS -- plus the neighbors now building similar empires) and poor CONNECT is buried in the wilderness. 

Try to keep CONNECT separated at least three feet from other wireless devices.