The root failure to concentrate on is that the PC does not see a new USB connection being made when the board is plugged in, correct?
If so the problem is either software based on the PC (and it sounds like your tried everything possible for that) or a bad USB cable (you tried that) or a bad FTDI chip on the board. It has nothing to do with the 328p processor chip, the PC should 'see' the board connection even if the 328p is removed from it's socket, so reflashing the 328p is pointless at this time, a problem with the bootloader would not prevent the PC from 'seeing' the board when first plugged in. Nor does the reset button have anything to do with establishing a USB connection to the board, pressed down continously, not pressed, pressed for just a second, all that wouldn't cause or prevent a USB connection from being established. I assume you tried to install the board to a different USB port on the PC? If not, try that.
The FTDI chip is not reflashable, it is a remove and replace component and because it's a SMD packaged chip, very difficult for most of us to unsolder and replace.
So the things you have done to try and solve the problem make sense, but I don't think that there is anything else you can try. The FTDI chip being 'quite warm' is also a sign that it is the root cause of the failure. The FTDI chip appears to be 'tits up', sorry.
That said, is it still possible to flash the board with the blown FTDI chip, but using another Arduino as an ISP? Or is it junk now?
Yes, it is possible that the board could be used that way, if the only problem is the failed FTDI chip. It's also possible to use a USB TTL serial cable to upload sketches from the arduino IDE. The only cavet is with the FTDI chip not functioning is impossible to say if there are any other hardware/firmware failures on the board additionally. The only common method of toasting a FTDI is by somehow applying voltage above the safe amount, and that could very well damage more then just the FTDI.