Greetings Patroclus,
Is it just a hollow tip with a rubber bulb attached? When the joint is melted you release the bulb and it displaces the molten solder? We had one of these in tradeschool but I never used it, maybe it was said that a soldersucker and iron was just as good. When the component is removed, you wait until it is cool, replace the soldering bit then wait for it to reheat before soldering the new component in.
Next there are vacuum desoldering tools, they have a benchtop vacuum pump and a weird looking tool with many filters. These typically cost well over $1000 and you will spend more time disassembling, cleaning or replacing the filters to keep the suction working, than using them.
Then there are plunger type desoldering tools, you heat the joint and when molten press the switch and an electromagnet pulls the plunger up displacing the solder. Some of these do not have replacable filters, just a chamber for collecting the molten solder. You just remove the lid, invert the tool over a bin, and the waste solder falls out! Well if you leave it too long between cleans the solder blob grows too big and you have to break it up with some small cutters or pointy nose pliers. As with any iron, after doing several joints the tip temperature drops and you have to wait for it to reheat before continuing. They are good for the larger joints but can damage tracks on small IC's. I think these are one of the fastest options as your iron can be ready to solder the replacement. These cost around $400 upwards
Lastly there is desoldering braid. I tried this when I was a school hobbiest, it got hot, burnt my fingers, would not flow much solder away from the joint and was green in colour. I had never used it since and is why I have spent so much in discovering the ultimate desoldering method. I bought it from a certain mail order company in Perth.
Recently I needed a better way to desolder IC's and have bought some desoldering braid from a reputable industry supplier and looking back can now see that my first roll was corroded or tarnished. This stuff is magic! Although it may seem expensive first, I believe it is the best way to go. You can get different widths for different sized joints. Don't use braid that is too small.
Samcheetah has the right idea for removing lots of solder, if you have access to a desoldering tool remove the solder with this, then finish off with braid. There are still a few people in the electronics industry that still use a soldersucker like you have. Back to your tip, you still have the problem of changing tips between soldering and desoldering unless you have a second iron. Is it worth the expence? Only you can decide, maybe you could try it, then you can tell us how good it is.
I do not know what Zevon8's soldapult is.
I hope this is of some help.