Problem with a lot of these desoldering tools is they are not temperature controlled and can't generate enough power to maintain temperature and keep dwell time low. Big risk on PCBs like that one if you use the wrong tools.
If you look at the TDS210 repair I did in the following picture, that was a Metcal and braid job with a subsequent clean up with isopropyl. The connectors are really really hard work as they are massive heatsinks on a ground plane but the right tools make it easy. I doubt a sucker gun could maintain enough energy in the joint without either cooling off and blocking or overheating the board and damaging it.
ChipQuick is an option but we were never allowed to use it for rework as if you don't clean all of it up (which is difficult) it alloys with whatever is left and can cause cracked joints.
Best thing for temperature controlled suckers is hammering through DIP packages. But I hate doing that anyway