I'm not 100% sure what the 'best' is, but based on my experience:
I think the only desoldering tool that really works are the fairly big/expensive ones that look like a gun and hook up to a vacuum source (either a built-in vacuum or a venturi system on an air compressor) like the Metcal 1351. I bought a Yihua 948D on Prime Day and it just doesn't work well enough to bother.
Best bet is a big chisel tip or hot air or one of the 'bodge a copper wire against all of the pins' methods, so that you can heat up everything at once and remove the whole component. Otherwise the best way is pin-at-a-time with pliers or tweezers, and making sure that the joint is nice and hot so that the solder on both sides is molten -- if you force it at all, you risk tearing out the pads. I have a JBC [15mm 'blade'](https://www.tequipment.net/JBC/C245752/) style cartridge which can take out pots and SPDT switches (but not dual gang pots or DPDT switches).
Desoldering wick for trying to remove a big component is desperation mode. In my experience, it works well for cleaning up holes once the component has been vacated, but isn't really that good at removing all of the solder around a seated component's pin. It only takes a little bridge of leftover solder to grip the component and that can be really hard to remove the last little bit with wick. It's actually kind of contradictory, the more solder you remove, the harder it is to heat the joint because the liquid solder transmits heat way better -- think of it like using cooking oil vs trying to cook something in a dry pan.
Sometimes adding low-temp solder (bismuth based / 'chipquick') can help because the low melting point of the solder gives you more working time to get everything hot all at once. (The tin/bismuth alloy solder melts at low temps -- ~140C v ~205C for typical leadfree alloys). This can give you enough working time for the soldering wick to draw everything in, or for the little hand-triggered vacuum desolder pumps to get in place and trigger. I've done this to a point by adding leaded solder while desoldering, which is ~180C melting point.
Flux helps the wick work better, because flux is solder magic. Flux really helps solder to flow properly, so it increases the capillary action of the wick quite a bit. A lot of wick will have flux built-in, but some extra flux paste can help things along.