Ahh nevermind. Figured this one out via brute force. Press and hold near the bottom of the trigger button—to keep the springs from flying away—then use a prying tool (I used a flat pick) to separate the top of the button from the midframe. The trigger button is held in place from behind with two circular indentations near the top of the button. Pry the button straight away from the midframe to release them.

The rubber cap on the joystick is slightly larger than the opening in the black gasket in the controller housing, so it will deform momentarily as you're removing the joystick.

Try to pull the joystick out at an angle, removing the side furthest away from the charging rail first. There is a small break in the black gasket that allows it to move out of the way on that side. The gasket is solid on the side closest to the charging rail, and there are very small wires routed over it that you will need to put back into place if the gasket slips over them while you're removing the joystick.

So again, try to pull the joystick out through the opening at an angle, so that the part of the stick closest to the charging rail comes out last. This will give you the best chance of leaving the gasket placed correctly.

Just another comment to echo what folks above are saying: You absolutely positively do not need to disconnect the ZR ribbon to complete a joystick replacement, and the warning about this should be given before the instructions on disconnecting it. There is very little slack in the cable, but it's just enough that you can rotate the midframe holding the ZR button out of the way without putting excessive strain on the anything. Just work slowly and carefully, and you'll be fine.

If you honestly didn't have any problems reconnecting the ZR ribbon, that's great and we're all happy for you, but you're clearly the exception.

Lift the midframe straight up, away from the motherboard, and keep a close eye on the ZR ribbon cable. Try—carefully—rotating the midframe so that the ribbon is not wrapped around the edge, then you can flip the midframe over and out of the way of the rest of the Joy-Con. This will give you a little more room to work without putting stress on the ribbon and without having to disconnect the ZR ribbon as described in the next step. You do not want to disconnect that ribbon unless you absolutely have to. It will suck to plug the connector back in later.