MSI did indeed have problems with electrolytic capacitors failing on their boards, but I don't think that's the issue here.
Thing is, failed/failing electrolytic capacitors will perform the worst when they are cold. The fact that both of you here have boards that boot fine when cold but not when hot eliminates the caps being the issue almost right away (though I will note that no one should ignore their capacitors if they have an older MSI board - the area around the CPU usually uses good capacitor brands, but elsewhere on the board MSI usually uses Teapo or OST... unless it's a newer board as is the case with O/P's, which can be seen to have Panasonic caps, which are of excellent quality... so that further reaffirms my notion above that the caps are not the issue here.)
Whenever I hear someone mention a board is "picky" about the RAM, especially if it appears to randomly work with the same sticks of RAM when re-arranged, the first thing that comes to my mind is some kind of communication issue (i.e. hardware fault) between the RAM and the memory controller... and most likely being caused by bad BGA and/or failed chipset.
For Intel -based boards prior to the 2nd gen i-series, the RAM controller was located in the Northbridge. Therefore, the fault could be in any of the following:
- bad contacts in the RAM slots
- bad BGA under the Northbridge
- failing Northbridge (or rather memory controller in the NB)
For AMD going forward with socket 754, the memory controller was moved onto the CPU. So for AMD boards that are "picky" or "sensitive" about the RAM, the following are more likely to be the cause:
- bad contacts in the RAM slots
- bad BGA under the CPU socket (certain OEM stock cooler retention brackets tend to put too much pressure, warping the board around the CPU and causing BGA failures over time)
- failing memory controller in the CPU (very rare, but I do have a confirmed case of this.)
My suggestion would be to start with cleaning the contacts in the RAM slots. IPA + cleaning with a brush, along with a number of RAM insertion cycles should be good enough for that. But if this doesn't resolve the issue... well the next most likely candidate to cause these kind of "random" faults is actually the BGA under the CPU socket, even though this isn't directly in the path between the memory controller and RAM. The reason I mention this is because OEM LGA775 coolers (especially of the push-pin type) are notorious for warping the motherboard and causing latent failures. It's actually surreal how often this happens - at least in my experience dealing with old 2nd-hand parts. So with that said, if you got the board to boot with a certain RAM configuration once but not again in another boot attempt, my suggestion would be to try and run the board on a flat surface with the CPU cooler only sitting on the CPU (with thermal compound, of course), but without being tightened down. In particular, if you get the board to boot, see if pressing on the CPU heatsink or Northbridge cooler makes the board crash all of a sudden. If yes, you've pretty much narrowed down the fault to either CPU socket BGA and/or NB BGA (with the NB BGA being many many times LESS likely to be the problem here.)
Lastly, and this is probably the worst news, is a failed/failing Northbridge. I know this sounds unlikely (and I also don't want to fearmonger about VIA chipsets), but I have seen several cases of VIA NB's failing - one of them on my own boards. The first time someone suggested this to me was a member on another forum who was trying to troubleshoot a s939 board (I forgot which board and chipset exactly.) Then, about a few years later, I had artifacts in BIOS and prior to the OS loading on my ASUS K8V-SE Deluxe motherboard (VIA K8T800 / VT8383). Furthermore, the artifacts disappeared once the system warmed up. Generally such symptoms would suggest a GPU failure... and that's all I though of it too (my video card at the time of this first happening was an XFX GeForce 6800 XT AGP that I got 2nd hand in not working condition). But then, I replaced the GPU with a GeForce 7600 GS AGP that I have used in other systems and knew was 100% working. When the same artifacts appeared on a cold boot again, but then didn't once the system warmed up, I knew something was not quite right. I tried playing around with various AGP settings in the BIOS, but none of it help... and I knew that would be the case too, because I had used both of these cards in this same exact motherboard without this issue. To reaffirm I wasn't just imagining things, I tried a trusty plain Radeon 9600 video card. When I saw the graphical artifacts in BIOS on a cold boot with that, the coffin was sealed: the chipset on my K8V-SE Deluxe is failing. No further questions.
With that said, it seems that nVidia chipsets aren't the only ones with problems (that is, nForce 4 series and newer.) Looks like VIA also might have produced unreliable chipsets probably starting around the Athlon 64 s754 era. Of course, compared to nVidia, the failures seem to be a lot more rare (or maybe more of them got dismissed as quirky "VIA doing VIA things" issues??) On the same note, I have also seen failing Ali chipsets from that era. So maybe everyone got affected during that era. The only two manufacturers I have nothing to say about is SiS and Intel. For SiS, I just haven't gotten that many board with them, so I simply don't have a large enough pool of data. As for Intel, if there is one thing they did right in that era, it was their chipset reliability: anything from the i8xx and i9xx generation is tougher than a hard rock.
Anyways, I'm getting carried away about talking about chipset reliability here... but in the grand scheme of things, I think this is where the O/P's motherboard might be headed. Well, either that or bad BGA under the CPU socket, since those are also rather frequent failures on LGA 775 boards. The last thing to try if nothing seems to make the board work better and if you're thinking about e-wasting it: reflow/re-heat. First try the NB, since that is usually easier and safer. If that doesn't do it, you can try the CPU socket (though these tend to be harder to do, mostly because the CPU socket has to be stripped of the metal parts, taped up on top with kapton tape to avoid melting it... and overall requires quite a bit more heat both on the top side and bottom side to assure the board doesn't warp from temperature differences.) Or alternatively to e-waste: chuck it on eBay "for parts or repair" or on whatever local classifieds website you like to use. I personally keep all of my bad boards for parts donors. The caps and the MOSFETs are typically the first to get used up, though I also occasionally dig in for various MLCCs whenever I get a video card with a bunch of them missing on its back (and I get those quite often.)