The technique shown in the video seems like a reasonable solution however you have a few more problems to contend with since it's a multi layer board. You won't be able to easily solder the jack to the traces directly under it since there is no longer plating inside the hole for the solder to wick up from the other side. Ditto with any inner layer connections that go to the jack pins. Pray that there is enough via stitching that it still works being only being soldered to the tracks on one side of the board. Being only soldered to one side of the board will make it significantly mechanically weaker also.
Charred PCB material is conductive so that would be the primary concern for a short. Usually you can try scraping any charring on the sufaces and drilling the holes slightly larger to remove the charred PCB material, but you run the risk of hittin the inner copper layers and pinching them together, creating a short. Given that the holes are slots instead of circular, i would just clean them with alcohol, scrape any blackened areas between traces and cross fingers.
I would use at least 20Gauge wire.
Oh and use slow setting epoxy (the stuff that takes 8hours+ to set) instead of hot melt glue. Glue it only after you have confirmed that the repair works so you have a chance of reworking it.