When using a braid, it sometimes helps to apply more solder, especially if it's plugged in a hole. Fill the pad, then use a clean section of braid that is still soft. Braids are flux infused, so once it's melted it's really not good anymore unless it's just a lump of solder you need to remove. The solder will come out quick. The trick is getting it to flow into the braid first, which is what filling the pad does, and being in liquid form, solder follows the flow, once the flow begins (surface tension and fluid dynamics mumbo jumbo). Sometimes more is less, as opposed to less is more. It'll be much faster than trying to heat the braid and get a section of it into a through hole.
I don't know if it's recommended for the case being discussed, but you can make a solder bridge and pull out the diode quickly. If you don't want a solder bridge you can build up the solder on the underside so that it's nice and tall, and stick a diode leg or a few resistor legs between the pads (elevated from the board), shorting out the component. Then heat the legs in the middle, and it should melt both pads simultaneously.
I'd prefer to just cut it, and remove a pin at a time, unless there's some reason you need to keep the diode functional. Diodes are thicker than resistors, so it should be fairly easy to get a snipper in there and cut out the diode.
Another possible suggestion is to get some thin bare metal wire, like used for picture hanging, etc. Loop it around one leg of the component. Heat the pad from the under side and then yank the wire. I've never known a component to be completely rigid and the component will bend on the leg that's still soldered down, pulling the heated leg up.
The wire trick is used, sometimes, for surface mount components (QFN), but instead of pulling up, you pull outward to get a slight gap between the pads and the component.