External wifi router can't to get internet through cable modem, but can when passed through another router

Reading what you've posted, it sounds like a simple case of the modem was not rebooted when connecting it to a different Ethernet device.  The modem will only provide a connection to whatever device was connected to the Ethernet port when it last booted up.  If you connect a different PC/router/etc., rebooting your modem should make it work again.

Edit:

To add to my above post, if you're DPQ3212 has the battery backup installed (which as old as they are you may have actually gotten a battery backup with it), that'll also need to be pulled during a power cycle otherwise the modem won't actually lose power when the power cord is removed.

There's a reset button on the back, all they have to do is press and hold it for 10 seconds. (Maybe less, but I like to be certain).