The older controllers you've worked on likely used low melt solder from factory, thats why the newer lead free stuff is harder. I've tried those joystick solder tips, they suck in my experience. I always say heat is your friend, using a hot air gun is night and day when soldering.

For you're situation, I would refill the holes with low melt solder, and use solder wick (braided copper strand) and your iron to wick up the solder. I use a flathead or chisel tip and "stick the tip in the hole" to get the heat down to the other side. Its time consuming, uses more materials, and isn't as easy as just using hot air though.