upvote
A desoldering pump (manual model, $10 or so for a decent one) is very suitable for removing solder from through-holes, if that is the main issue.
reply
The answer to almost every question in soldering is 'more flux'. Solder wick has flux in the center of the braid, but it's hard to get it into tight places like structural through-holes. Adding your own liquid/paste flux will make the wick much more effective.
reply
I often add solder to make it easier for the wick to get everything. If the original assy was Lead-Free, using low temp solder (I can has lead? As a treat?) may make a difference here as well. Flux pen on the solder wick also seems to help especially if your wick is kinda crusty.
reply
Melt the solder and thwack the board on something hard? So the board stops but the molten solder doesn't.

Sometimes though you just have to pile on solder and flux because the via is small enough that surface tension and heat dissipation means its never coming out

reply
Doesn't a pump make quick work of this?
reply
Frequently not. It's always handy to know about extra techniques in soldering.

You can also scale this up in a solder oven and remove almost every single component. Used this for reversing a PCB a few times.

reply
A good solder sucker is your friend.

I have the SS-02, and I like it - I had one of the cheap blue ones first, but the pliable rubber tip really makes a difference. If you’re soldering smd by hand, it’s more than worth the $20

reply