How do I make my FPS character move smoothly on a rigidbody platform?

Hi, I'm very new to unity. This is my first time asking a question myself instead of searching for the solution on the internet because I've been trying to do so for half a week now without the desired result.

So I wanted to make a game with hot air balloons. That means I need my player and item rigidbodies to "stick" to the basket of the balloon when it takes off. And the basket, as well as the balloon which is connected to the basket with spring joints, need to be affected by physics. Therefore both of these objects either need to be children of rigidbodies, or be rigidbodies themselves.

I have a rigidbody basket that parents other rigidbodies and the player when they step on it (repurposed from this: https://www.youtube.com/watch?v=rO19dA2jksk). This setup works just fine when the basket lifts "cargo" objects which are just rigidbody boxes, as well as a "rolling ball" player entity (which has a rigidbody component as well).

The problem probably lies in a simplistic first person controller I repurposed form this video: https://www.youtube.com/watch?v=_QajrabyTJc .

I can board and fly in the basket almost normally except my camera (or the whole player entity) actively jitters. Here's a video of my project demonstrating the problem: https://www.youtube.com/watch?v=ng80H8IRW5s

I suspect the issue lies within my first person controller since it is the only non-rigidbody that can ever board the balloon, as well as the only item with this problem. The wooden crate rigidbody is doing just fine, as seen in the video.

Can anyone please help me modify my first person controller in such a way to ensure the player has the smoothest experience flying the balloon? I've been struggling with this problem for almost a week now. I can provide all the scripts used in this interaction if needed.

Thank you in advance.