"The only "good" way to fix it is to reflow the chips. Anything else is temporary."
Irrelevant to the problem at hand. No error lights are present.
"It did display video and sound at first albeit pretty lousy until it went out completely ... It boots up normally and does not display any errors or red lights."
Bad chip solder doesn't noticeably degrade the experience. The 360 works one day, then is 100% broken the next day. Degrading video leads my first suggestion to be to swap out the video cable. If you are running the composite cables that came with it, then order a cheap VGA or component cable off Amazon/Monoprice. If it is new enough to have HDMI on it, then use whatever video port you aren't already using. Using HDMI? Then plug in a component cable to the analog AV output instead. If none of this works, then you likely have a bent video pin or an AV port that got bent and hit to hard so it broke some solder points. See if putting pressure from multiple angles on the AV plug give you a picture for a second.