I just have a couple theories on why the old SONY PS3's overheat and how I personally think to help keep them cool
1: Most people play PS2 games in HDMI in which I think is a main issue with the melting of some of the circuitry on some of the older models, my suggestion is to play PS2 games in AV mode with AV cables(aka) PS1 PSone and PS2 hookups only to reduce internal heat build up. After all when the PS2 came out their was not an HDMI hook up.
2: You should get a fan for them, they are rare but find able
3: The PS3 backwards compatibles need more heat sink chips in the circuitry in order to keep the machine from over heating on the insides.
4: It might be time to clean your system.
The PlayStation 3 has been around for years. There isn't much of a need to theorize what's already been confirmed about the hardware forever ago.
1. Video output has nothing to do with the "melting circuitry" of the system. One of the primary contributors to the original model PS3's absurd temperatures were the "ZSSR5391A" model and "APS-226" model power supplies. These power supplies were found in launch model PS3s, and were later replaced with the "APS-231" model power supply, which consumes significantly less power and thus runs much cooler. The "APS-231" can replace any of the two older power supplies with no technical issues. It is highly recommended if you want a cooler running system.
2. As far as a fan goes for the system, I'm hoping you're not referring to an external fan. There were some PS3 owners who purchased Nyko brand external fans that were powered by the USB slots of the original model PS3. According to these users, the power draw from those fans only produced a greater amount of heat, and supposedly increased console temperatures by a fair amount.
The fan you should look to replace is the PS3's internal one. There are two models: a 15-Bladed fan and a 19-Bladed fan. The 15-Bladed fan is known for having a poor design and circulates hot air back into the heat sink. The 19-Bladed fan corrects this design issue and is much quieter in comparison. The fans are not rare or difficult to find. You can easily purchase one for about $20 on eBay.
3. You're correct on this somewhat, assuming I've deciphered this correctly. There is no "Heat Sink Chip" though. Both the GPU and CPU of the original model PS3 came with standard factory thermal paste underneath both heat sinks, and produce quite a bit of heat. The majority of repair tech gurus recommend replacing the stock thermal paste with Arctic Silver 5 or some equivalent, and even the heat sinks with more durable ones. There's also the matter of the poor lead-free solder used to attach the GPU and CPU to the motherboard, which many recommend replacing with lead solder to avoid the Yellow Light of Death.
If you're feeling particularly nuts and want further preventative overheating, you can follow the example of a few people who water-cooled their PS3s instead, but let's not get into that.
4. Yeah. I think any time is a good time to clean your system. A little long winded, but I couldn't help but correct some of the misconceptions here as I own a backwards compatible PS3 which is still fully functional to this date. I've done nearly all of the above to the system
(minus the water-cooling), and have zero issues with it. I'm sure it'll die someday as all electronics do, but the thing's been with me for 9+ years so I plan to squeeze as much life out of it as I can.