How old is the motherboard? I ask because if the CMOS battery is running low, then the BIOS itself can make the system unreliable. In that case a new battery and BIOS set to sane defaults would solve this.
Similar for heat issues. If the heat sink compound is one which solidifies and loses effectiveness over several years, then you might solve a heat issue just by using new heat sink compound on the CPU.
It isn't likely to be a big help, but sometimes reseating any PCIe cards or connectors to the motherboard and other peripherals can help. Connectors which stay in place for many years can suffer from thermal creep (heating and cooling cycles over many years can cause the connector to actually move slightly and cause connectors to fail just like when not fully seated).
You could also test with memtest86+ for a day or so. Typically memtext86 can verify basic function. If memtest86 does not fail, then you at least know basic RAM and CPU function is in good shape (without hardware for that part of the system being at fault).