Adding an Ethernet Switch to my Airport Extreme

We finally got started on an Original AAAD project last night - wired Ethernet.

I've been thinking about this one for years. 

And in the interim, wi-fi has improved to the point that it's almost a pointless exercise. 

But I'm still hopeful that a Gigabit Ethernet network will be more reliable for our Apple TVs.  

So, when we were finishing the basement, we had the electrician run Ethernet cable to our TVs and the home office. 

And we acquired this Netgear GS316! It's an unmanaged switch, so it should be plug-and-play. 

So far, that isn't my experience. 

We have the following network configuration (it's extremely basic):

This really, really should work. (Wi-Fi devices aren't represented)

Bottom line - the Airport Extreme doesn't seem to be talking to the switch. 

The Cable Modem is plugged into the Airport Extreme's WAN port. 

The Switch is plugged into the Airport Extreme's first LAN port. (The Ooma and ADT pulse, while not pictured, are plugged into the other 2 Airport Extreme LAN ports.)

The LAN port going to the switch is NOT lit up. 

The Apple TVs report that their Network Status is "Ethernet - Connected", but also report NO network or internet connection. 

If I had to guess, I'd say that the AppleTV sees the switch and recognizes that it's on a network, but that network doesn't include the Airport Extreme or anything else upstream. 

I'm hopeful that by powering down the entire network, then powering up the cable modem, Airport Extreme, and Netgear switch (in that order), we'll be able to get things working. 

To get started, I'll bring all of the gear upstairs, so I have it all in the same room. 

I'm skeptical, though. This should have worked last night!

Does anyone out there have any experience in getting an Airport Extreme to talk to a Netgear switch? Any idea what I could be missing here?