You're either setting the iron temps too hot, or leaving it on there too long. I can tell because the flux around the area is burnt, and some plastic pegs are melted.
In the last pic, it looks like the traces on the "Y" and maybe the "H" keys are severed. Can't tell if any other switches are damaged from the blurry pics. Can you point out exactly which keys are not working?
And like some people mentioned, some pins need more solder. You need enough solder to come out of the hole, but not too much where it starts to bulge up. A few pins, such as the one next to D42 (first pic) still has a gap in the solder joint.
Those are to channel solder through the hole to the other side of the PCB, for greater support (so each pin is more "solidly" in the PCB). Filco advertises that in their product ads as a feature. And it also makes desoldering switches from Filco PCBs a greater annoyance. I freaked out as well when I pulled a ring out, but turned out it was no big deal. The metal ring on the outside of the PCB, where you're soldering, is the relevant part to make a metal-to-metal connection with solder, as that's what is connected to the traces.
Although you're right that those tiny "cylinders" are not needed, it should be a big deal that you're pulling them out. That means you're not desoldering correctly, and pulling them out is physical damage to the solder pad. The two pads on both sides along with these cylinders are called "plated though-hole", and that cylinder connects both sides of the pads together. If you pull it out, you risk pulling the pad too. I just wanted to correct this so newbs won't think it's perfectly ok to just rip these out.
P.S.: Oh I see, you still have the plastic pins for PCB mounting left on. I thought they should be clipped when used with a plate. That could be an issue. Look up plate-mounted switches vs. pcb-mounted switches.
Regardless if you're using a plate or not, if the PCB has holes for the two plastic pegs on the switches, then you should use PCB-mounted switches. Using a plate AND those two pegs makes a very solid mount. The only time you're forced to clip the two legs off is when you're using a plate and there are no holes in the PCB to use PCB-mounted switches (like Phantom, Filco, CM, etc).
This is why we only ordered PCB-mounted switches in the GH60 GB, even though almost everyone ordered a plate.