That seems incredibly stupid.... and raw mercury may actually be harder to find and more expensive in the US than a cheap hot air gun.... and while the liquid may not be dangerous to have around, it's very dangerous to handle, so the residue on the iron, the splatters from flux boiling, the residue on the board.... all of it would be effectively hazardous waste not suitable for a landfill.
Mercury's boiling point is about 350C, too, so it doesn't take much extra heat from your iron before you're breathing it even with a decent flow fume hood. They overlay "safe soldering" on the video like they're trying to reassure themselves...... the right equipment is so inexpensive, don't throw your health away handling and trying to solder with mercury.