Well I did eventually figure it out after screwing around for another few hours.
It turns out the video you linked earlier is slightly out of date for the newer cars. I already mentioned that the screws were different, but it turns out there's also an activation pin in the ignition switch that has to be pressed in. The screwdriver I was using to turn the switch didn't press the pin in all the way. Without that pin pressed in all the way, turning the switch to the 'start' position does nothing.
In addition, removing the screwdriver and thus releasing the activation pin after the switch has been turned to any position past 0 (and especially after the car has been started) causes a lot of the electronics to go haywire. All the dash warning lights come on, the windows can no longer be opened or closed, the airbag gets messed up, the antiskid gets messed up, etc.
I found elsewhere in another forum that there's a metal plate on the bottom of the tumbler that you can cut off (with serious effort) and perhaps release whatever's broken inside. I eventually did this and after tearing it off and pulling out a spring the tumbler is freed and I was able to reattach the switch to the tumbler so I can drive it to the shop. I didn't solve the airbag and antiskid issues though, I guess I'll have to pay the dealer to reset those.
If you happen to have a couple hours and a dremel with a diamond cutoff wheel, I'd recommend cutting the panel off first and see if you can't fix it that way rather than doing the screwdriver method. It's SIGNIFICANTLY more work, but you'll avoid the pin/electronics issue entirely.