(Summary at bottom)
I've spent the past week trying to get someone to play Portal 2 co-op with. My brother said he'd play on Sunday but we couldn't get it all working. I figured out it was that we needed a controller for the second player, and I was more than able to get one going, but he didn't want to play anymore after that. Today I finally got him to try again, and after a ton of troubleshooting and waiting and figuring things out, we've got the game on both our screens, and this time the option to play local split-screen co-op is there.
But he can't move his camera.
I try looking at everything, it's all set up right, it should be working, but no matter what the right stick won't register.
And.. at one point the left stick cut out too, and I tried swapping out for a different controller and that fixed the left stick (and swapping back later still had it working), but the right stick still did nothing.
And, well quite frankly that's it. I did everything I could, and at one point I thought i'd got it, when I looked up the issue and found something about Steam Controller Support messing with the right stick so I turned it off as it suggested, and it looked like it was working! When I loaded it up I could move the cursor using the right stick from my brother's computer, but when I opened into gameplay it still wasn't working.

So, I've done some testing and gathered some information on the details. I'll give as much information as possible, and hopefully someone can help me.
We're both running Steam in Big Picture mode.
We tried using two different Nintendo Switch Pro Controllers connected by USB C cables.
The right stick itself works, and it works on the computers too.
I know the problem isn't his computer because the right stick can control the cursor out of game.
I know the problem isn't Portal 2 itself because I tried playing with the controller on my computer and it worked fine.
The problem is definitely not in the bindings.
The right stick click input still registers fine.
I do not know if the problem is present when playing remote play on other games.
I do not know if the problem is present only on the client or if there are troubles when the controller is connected to the host as well.
The blue lights on the controllers that light up when connected to a PC were behaving abnormally, shining very dimly at times and turning off even though they normally persist for days in my experience (probably irrelevant)
When checking in Steam settings on the host during remote play, the client screen would display a message saying input was paused while the host was busy. I was still able to give input through the controller on the client, though.

Ultimately, the problem seems to be specifically in when playing the game through Remote Play, as the problem does not exist in any other conditions. I am absolutely at a loss at why this is happening and just want to play with my brother. Could really use some help.

TLDR: When trying to play Portal 2 Remote Play, inputs on the right stick do not register. The problem exists exclusively in-game during Remote Play session and I can't figure out for the life of me what could be causing it and how to fix it.