The broadband light on the Arris NVG510 is blinking red on an
Ethernet-connected Win10 desktop computer. Restarting the router
didn't help, and it continues blinking red even when the Ethernet
cable is disconnected. However, all smartphones on the network have
WiFi working and connected to the expected SSID. Why doesn't the
broadband issue affect the wireless connections?

If you meant the indicator on the nvg510 is blinking red, I found this:

Gateway status lights - NVG510

Broadband light Flashing Red
There is no DSL signal on the line. This is only used when there is no
signal, not during times of temporary no tone during the training sequence.

Thanks for your reply. Yes, I read that on page 70 of the Arris NVG510
router manual. The fact that there's no DSL signal while the WiFi is
still working has me befuddled.

Ethernet light Off
The gateway power is off, There are no powered devices connected to the
Ethernet ports.

https://www.att.com/esupport/article...M1050512?gsi=c

Can you access the modem logs? (log in and go to the "Diagnostics-Logs"
tab)

I was actually helping a non tech-savvy user over the phone, so I
don't have that info and there's no chance the user can follow
directions to get that info.

How old is the gateway?

They've had their router for about 3 years but they bought it used.

However, AT&T is coming over on Christmas to troubleshoot so I'm not
concerned about fixing anything anymore. Out of curiosity I'm
concerned as to why their WiFi is working while the Broadband light is
flashing red.