My 2 cents: yes, a desoldering gun is likely to make your desoldering much faster and much much easier, especially if you have only one good hand.
I have been using wick and solder suckers and a Metcal iron for a while and yes, desoldering lots of pins is possible, but it requires time, a very good technique and the occasional problem (solder splash, glue...) because, well, most of us do not have a perfect technique.
I recently got a good price on a Metcal DS-1 and oh-boy is this thing simple to use. Push head against solder, wait for solder to melt, press trigger, wiggle the pin, release trigger. Total time 3-5s per pin and a 100% clean and easy job. All with one hand no real difficulty. These things do clog, but mostly because of bad technique (i.e. releasing the trigger while solder is still being sucked in).
So, if you have the money, I would say that a desoldering gun is a good investment, especially in your case. I almost got a FR300 before finding the Metcal. It seems to be a great all-in-one gun that should work for you. If you're patient, you can get really good prices by waiting for a 25% or 30% sale on Zoro, otherwise get one at tequipment.net with the Eevblog discount.
The Metcal is great, but it requires a Metcal power supply and shop air (I use CO2, it's easier for occasional use), so unless you're ready to hunt EBay for good used ones (or want to switch your soldering station to a Metcal), the Hakko or its older brother the 808 are probably the best deal.