[Eliot Ness]

Tim, I would be leery of the pistol grip unit on eBay.  The trusted name in affordable desoldering stations is Hakko, but it will be 2-3x as much as the one in your link.  They are geared more for PCB work and I read where they tend to clog when doing big clumps like you mention.

For years I used solder wick and various small aluminum bodied solder suckers that worked OK.  Then a few years ago I splurged and bought a U.S. made Edsyn Soldapullt DS017:

http://www.edsyn.com/product/DS017.html

https://www.ebay.com/itm/Edsyn-desolderi...SwbF1aEku2

The DS017 works great and has readily available replacement tips and O rings.  I still use braid also, but the Edysn was one of the best $21 I've spent.