There are two alternative solutions.
Don't bother whicking the solder.
1. Add lots of 60/40 solder on both leads of the capacitor on the back side. Put solder iron on both blobs of solder at same time until solder is liquid and pull capacitor out.
Put new capacitor leads on the holes, warm up the solder on other side and slowly push the leads in. The blob of solder will get stuck to the lead and move away from the board and now you can put fresh solder at the base of the leads.
With this method, there's a small risk that when pulling out the capacitor some solder will get into the through hole and will make it harder to insert the new capacitors.
2. So here's the twist to this method. Get some pliers, grab the capacitor and gently twist it back and forth to weaken the leads, after about 20-30 small rotations/twists the leads will break and there's going to be a few mm of leads on the top side.
Now you can grab some tweezers, catch the lead in them, add a bit of 60/40 on the bottom, heat up the solder and use the tweezers to push the lead through the hole to the other side. Repeat for the other lead.
I did the first method a lot and in 10-20% of the cases I ended up having to use fine stainless steel to clear the hole for the leads (you can find sets of such needles on eBay quite cheaply, they look like they're home made but they work).
With the second method, it worked fine almost every time.. usually the lead breaks close to the capacitor leaving quite a lot of wire on the top for you to grab and push through once the solder is hot. And since the capacitor is no longer there to absorb heat, it takes less time to heat the bottom of the pcb and the solder, you notice quite fast that the lead is slightly moving around and you just have to push it through.
In very few cases, the lead breaks flat by the pcb so it's harder to push it, but it's really not a problem if you have some stainless steel (or something that doesn't stick solder to) to push the remainder of the lead in either direction and clear the hole.
edit: and btw, I'm using an old Hakko 936 with the standard conical tip, set at around 400-420c for motherboards (because the ground plane absorbs heat fast)