I just realized something as I was getting ready to try this today: Do I need one of the old Archer game switchboxes with an RF output to try this? If so, I have no idea what happened to it (I assume it was left hooked up to the TV when the TV was thrown out.) Do they still sell these things anywhere?

Yeah. Radio shack or frys should have them. Get one with a switch though. The automatic ones ( like the nes one) won't work with atari. You can also get an rca to f (coax) adapter and use a regular rca cable. They should have this at radio shack too.

edit:
not sure what you are asking about as you said you had an f to rca adapter already.

Well, I was going to try demodulating the Atari 7800 signal through a VCR, then hooking the VCR up with RCA cables to the TV to see if my TV would display the Atari signal that way. I'm just curious as to what ways I can hook up my old consoles and have them display properly on this particular TV, since it's so finicky about the signal through the composite hookup.

Connect the vcr with an rca cable. then connect the atari to the vcr the same way you were connecting it to the tv. you can also try to connect your snes and n64 games that werent working through rca inputs into the vcr and see if they work also.

Sorry for the delay, guys, but here's the verdict:

I first tried running my Atari 7800 through a standalone VCR that was connected to the TV via RCA cables. Unfortunately, this did not work, because the VCR itself elicited an "Unsupported Mode" error.

Next, I hooked up a DVD/VCR combo to the TV via RCA cables, and every system (NES, SNES, Atari 7800, and Nintendo 64) worked without exception. This is an odd TV lol.

Did you ever call them and ask about it? Its either not working like it supposed to or they built it with really poor quality components. My guess is its just cheaply made without much thought given to older hardware that would be connected to it.

I haven't called them yet, but I'm thinking along the same lines as you: cheaply built with no thought of 20-30 year-old video game systems. Plus, it was a Black Friday special, if that tells you anything LOL. I'll call them tomorrow and find out for sure.

OK, so I finally got ahold of Vizio, and talked to a very unhelpful douchenozzle named Nicholas, who kept telling me I had an output issue. I told him every system that doesn't work on my TV works on every other TV I've tried, so I know it's not an output issue. I told him to get a supervisor, and the supervisor actually knew what the hell he was doing. He confirmed that the 240p signal is not compatible with this model, but that I could get an upconverter or just hook it up through the coaxial.

Thanks for following up! Figured that was the case but wasn't sure if anyone at Vizio would actually know that or admit it if they did. Sucks that some of these newer TVs aren't compatable.

Well, I talked to the owner of my local game store, and he has a 32" Sony Trinitron tube HDTV he's just going to give me, so I'll have a good (although very bulky) gaming TV.

He's not giving it to you, he's trading it to you to pay for carrying it out. Bring a friend and dolly, those TVs can be heavy.

Haha I know; my older brother had a 36" that I got to help him move.

The game store owner did tell me if I picked it up from the store, he'd charge me $50 for it. If I pick it up from his house, it's free lol.

No problem, mickcris! I'm glad to know that it's a compatibility issue, and my TV isn't broken lol.