2005, HDMI was just being introduced, so can we assume you have the older composite feed? (Coax with RCA plugs.)
The DVD player is working, at least the audio is making it to the speakers.
Here's how I'd approach it.
Get a small portable TV for a test. Take a handy red/white/yellow cable and hook it between the DVD player and the portable TV. Do you see a picture? Make sure you're on the "Video" input.
If Yes, then hook that portable TV to the cable that was feeding the existing TV. (Normal up the DVD end) Do you still get a picture? If yes, then the issue is in your TV, if no than you have a problem in the cable. Cable problems are almost always in the connectors (assuming AS didn't put a screw or rivet through it.) Pick an end and cut it off, replace it with a new RCA and repeat. Since the TV swivels and the DVD player doesn't I'd suspect the TV end first.
As always, take one step at a time.
Oh, and I found soldering a RCA is a pain, so I'd just buy a RCA cable and cut the molded plug off and splice the wire.
This sounds weird, but you can "hear" video on an earbud. If you're tracing video, the sync burst sounds like static if you touch the wires to an earbud. Old Indian trick.