My advice would be to work on your equipment/technique a bit, then. Make sure your iron is decently grunty with a good sized tip in good condition. A common mistake is to keep using tips that are flogged and won’t wet out with solder cleanly, or using a much finer tip than you need. Changing the tip to the biggest one you can get away with is important. Having a range of tips available helps a lot. (Edit: Went and checked, my standard would be the T12-D12 which is a 1.2mm wide chisel tip)
Another important thing is making sure that your iron is actually the right temperature, especially if it’s a cheaper one. A standard K-type thermocouple should be all you need for that. I’d give the tip a good clean first to avoid depositing anything on the themocouple bead and maybe measure it just back from the tinned point. I use hotter temperatures for rework/disassembly than I do for assembly. Assembly: 300C for leaded/350C for lead-free. Rework/disassembly: 350C leaded/380-400C lead-free.
One trick is to consider what you’re trying to work on and what solder it likely has on it, plus what solder you have on hand. Most newly assembled stuff will be lead-free and often the cheapest high-tin content, high melting point form of that. This stuff melts hot enough that it burns off flux and oxidizes much faster, so reworking it usually ends up in really nasty, crystalline and ‘stringy/goopy’ solder. If you have leaded solder, adding that in significant quantities (2-3x what was there to start with) to the joint will lower the melting point significantly as well as adding more flux that will reduce some of those oxides and protect it from oxidizing further as you work. If you don’t have leaded solder, I’d strongly recommend getting some. It’s a single easy change to make your soldering work significantly better. If you can’t get leaded solder then at least get some lead-free suitable gel flux in a syringe. You could also use the the flux pens, I think they suck to use but they do work. That’s a distant 2nd to just adding leaded solder to the joints, though.
Removing the pin headers is usually super easy because you can pull the pins one at a time with pliers. It’s destructive but pin headers are pretty cheap. Doing it this way I wouldn’t even bother with a vacuum desoldering setup or using solder wick, I usually find them to be more trouble than it’s worth on this style of pin form anyway. The key thing here is to make sure that everything is hot and molten, especially the traces that connect to grounds/power pins, so you’re not pulling pads off the board or pulling out through-hole barrels. Not the end of the world to blue-wire a fix if you do, but it’s much better to avoid it altogether.
Once you’ve got the pins out of the holes, that’s where I’d lower the temp on the iron, add some fresh solder, reheat each through-hole and then wick/vaccuum/blow the solder out.
So, the process:
Set up your iron with a tip that’s suitable for the pins, well tinned and set to ~350C.
Put the board in some kind of vice so you have easy access to both sides and it’s well retained.
Add a decent sized blob of leaded solder to each joint. Do that for all accessible pins.
Then, for each pin you want to remove:
Grab the pin tightly with a pair of pliers but don’t pull yet
Apply the iron tip to the joint for longer than you think. This will depend on the iron but maybe 5-10 seconds.
Wiggle the pin with pliers while doing this and if it moves freely, wait another 3-4 seconds then pull it out. You want it to be WELL molten, not right at the edge.
Do that for all the remaining pins.
Once you’ve got that clear, I clean the board with a lot of isopropyl and a cotton swab/piece of cotton wrapped around a small screwdriver while it’s still warm to get rid of the flux residue.
Add a little more fresh solder to all the through-holes.
Run along them one at a time reflowing them and wicking out the excess solder. Moving quickly through this will help with the grounds being hot.
Give it another quick clean and you should be good to re-assemble.
If that took me more than 10-15 minutes, I’d be surprised.
Edit:
Out of interest, my equipment for that would be:
Hakko FX-950 w/ T12-D12 tip
Chip-quick lead-free tip refresher paste
Chemtronics desoldering wick in a few sizes
RS Pro branded 63/37 leaded solder
MG Chemicals gel flux
Wiha micro-pliers
MG Chemicals double-headed cotton swabs
Menda Isopropyl pump bottle