So as I confessed mere hours ago, I bought myself a PS3. Apparently God isn't big on oathbreakers, as my experience thus far has been mediocre at best.
Long story short, it turns out that having the PS3 connected to the TV messes up any signals going in on any inputs except one. So if I want to be using the PS3, I have to have it connected via HDMI or it messes up its own signal, and if I want to use anything else, I have to have it disconnected completely.
That's fantastic. I've never heard of anything like that before. I've tried everything I can think of (and I can think of a lot), including exchanging the system for a new one, and that's the only conclusion I came up with. Bear in mind that my TV is Sony too.
I made the mistake of calling the Sony support line. The supervisor there told me that my best bet would be to either live with it, or replace my old, probably broken TV, with one of their fine Bravia LCDs. I told them that since I just spent $400 on this new PS3, I can't afford to buy one of their fine TVs, but if they wanted to give me a $400 discount, I might be persuaded. Unsurprisingly, they refused.
They also offered to let me send it to them for repair. That was nice of them I thought.
So, my faith in Sony's engineering remains very low.