I agree with Kleinstein, the whole thing needs to be sensitive but is way too sensitive lol.
I would first suspect flux contamination in/under the jack and don't forget the other PCB traces nearby. The factory PCB wash can have troubles with long parts or ones missing a vent hole, and left some residue. I find all fluxes are hygroscopic.
You can just pull off some of the cotton at the end of the q-tip to fit it inside a jack to clean it. Most I find are really dirty inside.
I would've followed the PCB trace to whatever the pullup resistor/protection clamp/logic gate is, to know what the circuit is.
It would be interesting to test the firmware's behaviour when there is something detected in BOTH jacks at the same time, that could cause a chuckle.
As I understand, the 5MEG resistors are huge 2W parts (3.2kV) because HV can only ever appear right at the jack when the fuse is blown. The UL fuse clear test is min. 3kV as I remember, they add inductance to make an arc happen.
For the Soldapullt, technique I was taught was to never let the tip touch the soldering iron, it will melt it.
You heat up the solder joint, fast like lightning pull the soldering iron out of the way and then put the Soldapullt nozzle on the liquid solder, square on the pad and hit the button.
You get better sealing and suction this way. Also, the piston really likes some light grease Lubriplate on it.
I know all this because I was poor as a tech, no fancy-schmancy desoldering stations. Used that desoldering tool constantly.
Lately the old Hakko I have is shit, lead-free clogs it up easily. I spend more time cleaning it.