Man i really wish i could see a video. Though i totally understand what your explaining, maybe it would give me more confidence. I would much prefer to just squeeze the pins and separate, then resolder when i put it back together.
If I have time later this afternoon I'll see if I can dig up something around here to video it. I think I have some backplane-plugin cards for some telco equipment where the connectors on the boards all use those press-fit pins instead of being soldered. Those springy-pins are actually quite common.
Can you not get any area of the pins to even
start shoving back up through the board by just pushing on the bottom of the springy pins, trying to push them up through the board? Sometimes I can just push down against the workbench / table or something (maybe the end of a screwdriver handle?) and work across it from the bottom. Works well with connectors where you can't pry from the top in any way until you create some room between the PCB and the connector housing on the top side, so you HAVE to start by pushing the springy pins back UP through the board before you can even try wiggling and prying the connector from the top.
I've been fighting a nasty cold and don't have much of a voice right now but I could probably squeak out enough of an explanation into a video and test the "picture is worth a thousand words" axiom if I can find a piece to demo it on.

Also, you shouldn't have to solder them when you stick it back together. Those pins create a LOT of force outward on the PCB holes and even after being removed once or twice will still have plenty of pressure to make a good connection without soldering them. This makes it possible to get apart again later if necessary. If you solder the springy end and ever need to disconnect it again you will have to desolder the other regular PCB side as you will never get the solder out from the springy pin side. They're jammed in there WAY to tight. This is why you should NOT heat up the area to try to pull it apart. If there is any tiny bit of solder plating those holes, you will make your task MUCH more difficult by melting it onto the springy pins!

That WOULD work, of course, on your board-to-board situation but will NOT work on connectors that are springy-pin-press-fit to a PCB since there is no "other" side to desolder.

The only times I have ever seen bad connections on those springy-pin things it was from SEVERE corrosion, and they had never been removed before or anything. Your pictures look perfectly clean and in good shape. You just need to carefully pry the things apart and they'll just squeeze back together after your repairs.
Another thing that worries me about cutting the pins is that i'm going to lose structural integrity between the boards. I guess if it comes down to it, I can just replace the entire header
You COULD do lots of things but you'll be creating a lot of unnecessary work for yourself.
