What do you do if the structural through-holes already have solder in them, that wick doesn’t seem to get? I’ve been trying to put a new USB C port onto my switch for quite a while now. (Now that I think about it, I can probably just shorten the prongs on the port and add solder after for structural strength).

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

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.

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

Doesn't a pump make quick work of this?

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.

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.

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.

How would tinning those tiny pads not create a massive bridge between them? Does the bridge somehow go away in the reflow phase? (Not familiar with reflow at all)

To add to the sister comments, you can quite easily remove such bridges by adding flux and then touching each individual pad with a fine tipped soldering iron. It sometimes takes a few tries, but eventually the solder that’s touching the solder mask will either be wicked onto the iron or move onto one of the neighboring pads. (The trick is to touch just the pads with the iron, and not to try to attack the solder bridge itself.)

Yes, the surface tension of melted solder pulls the solder to just the pad areas (assuming you don’t have far too much)

Make sure there is soldermask between the pads. This makes soldering much easier!

(If your foundry can't fabricate it, then make the pads thinner until they can fabricate the soldermask.)

Using with a little flux while tinning usually prevents the pads from bridging