I would second that the best approach (if RMA will be no go) is to look into soldering that new connector. Although, I have not seen the board from the connector side, so it may be a challenge for most people, so we also agree here that this route may not be for everyone.
The second option is to cleanup that melted connector. However, we may have problems with ground pins. Yes, the first three pins can be cut off (all three are for 3.3V, so not needed at all) to not disturb the connection. Also, the orange cord should be gone (just in case, and don’t cut the red one, as that is 5V).
Kind of zooming on the second picture, it looks like there was a short between pins 3 (3.3V) and 4 (ground) (was there a tension on the cable?). So, pin 4 is a goner and should be removed. It is hard to say whether pin 5 (ground) is damaged, but most likely it is (melted plastic, not so much copper). It would be really good if pin 6 (ground) could be salvaged. That would leave the drive with 3 or maybe 4 ground pins (pin 11 is optional, so not sure whether it can be counted as working ground).
We can assume that if those 5V and 12V pins are duplicated, we need the same amount of ground pins, and we have 6 of those total, but only 3-4 ground pins. Although, I think that max 5V kicks in before max 12V is present (5v goes to the board and stepper, and max is at the boot; the board once settles down powers spindle, so this is the sequence we should see). So, maybe those 3-4 ground pins will do the job. Also, the max power time is rather short, so once running, the power drops by 2x-3x, giving more headroom on those ground pins.
Also, that power connector is designed for drives that potentially more power, plus has to have some headroom.
Still, there is a concern that those salvaged pins may have bias on the connector side, making the whole connector less reliable - more prone to plastic melting, but not really shorting.
Based on that, I would get the soldering iron first. If that is not an option, I would clean up those pins, and use it as is. I would for sure cut that 3.3V cable, and potentially test the PSU, whether it will be tripped on 3.3V shorts (as @Fuzeguy stated, PSU should act first and not let this short burn those pins).
By the way, I would also ask Toshiba (in case they refuse RMA it) whether they could provide the board or sell it. Toshiba is far behind Seagate and WD, so may consider that to make customer at least not fully disappointed.
I would also sit for some time on this drive (if not used), as the next Toshiba may have mechanical failures, as such can be the board donor to this one.