I tried that first, and many times with the TV on or off first relative to the PS3. The PS3 shuts off right after the second beep - about 5 seconds - even if you remove your finger from the power switch very quickly - the button turns red and the system powers down. This changes nothing on my HDTV. Looking at the menu on my SDTV, I see that it is mostly at default settings (i.e. No HDMI). On my TV which does not support 1080i, can I hit 'X' twice after selecting the 1080i resolution and then exit keeping that setting?
It is supposed to power down quickly like that. You are doing a video reset of the PS3, which it takes it down to basic video levels. It's designed for the situation where you maybe had the PS3 plugged into a modern 1080p television set for 1080p output, and then you move the PS3 to some older more basic TV that can't accept the signal.
That's very much your case I believe as your 55 inch RPTV is, sad to say this, pretty old.
TV's of that vintage didn't support many of these higher modes. And sometimes, even if a TV like that did at one time accept say 720p, it would be at the borderline of what the circuit boards could accept, and over time their ability to process the higher bandwidth was sketchy.
Anyway I think the reset takes it down to 480i or 480p and clears the check boxes for those higher modes.
Can you hook up the composite connection to troubleshoot this? I believe you are calling that 'SD' right? (NTSC is not an relevant here) You may know this already but composite is the single wire video, usually carried over a cable with yellow ends and typically called 'AV' input on a lot of TV's.
I would recommend booting up and viewing the PS3 XMB menu screen over your composite video. Using that, then change your component display settings down to the lowest possible, 480i.
Triple check your wiring to make sure you are indeed using red-red blue-blue and green-green. I have seen dozens upon dozens of cases where people mixed the blue and green wires because they can look similar in dim light.
If you can get 480i to work, they progressively test better display resolutions in that XBM display test function that it sounds like you are familiar with.
If 480i doesn't work, try 480p. Also try the component 1 vs component 2. Old TV's like that sometimes only had the 'good' circuit attached to one of the inputs, meaning that one of them can accept the higher progressive signal and the other sometimes can't.