You're supposed to get the solder pump very close, it can even touch the soldering iron, it doesn't melt significantly to cause issues, if you watch some online videos, some people just push the desolder pump onto the iron
I have 3 soldering pumps, one is the cheapest kind, one is a good plastic one, one is a great metal one, the good plastic one is the best, as it's easy to re-use/reload, all of them pump better than that, but I'm guessing it's because you either don't heat the solder well, or keep the desolder pump too far away
Ideally, you should touch the solder for a second, it should get completely liquid, and you should pump it out in 1-2 tries, it gets better with time (it's 4-5 tries with some lead's)
I also noticed that Weller lead is much better than others, it gets soldered and de-soldered easily, some cheap ones stick to the board and keep resisting, stock ones are usually like these
So I suggest also buying a quality solder lead with built-in flux