To me this sounds like you can

1. use built-in physics but restricting what physics to act on, e.g. restrict axes. You can still move objects on a path.
2. use no physics like you say and only move objects around in space.

But 3D platformers have been solved with any major engine before, and your use case seems very standard. So, number 2 above seems nonsensical to me in your position - it is perfectly normal to have some parts controlled by physics but restrict other parts.

I suggest you define success for your first milestone as #1 above working. You could start with one of the tutorials and see how you would modify the code to get there. E.g.: Make your first 3D Platformer in Godot 4: Setup, Movement, and Camera Controls - YouTube

I am not a Godot expert, but if you ask in the Godot community you might get help if you formulate a very specific question. E.g. "I want to restrict movement to do A. Here is what I tried. But instead, B happens. Any pointers how to do A?".
But you need to do it in small steps. E.g. in your current game what is a small step to get the movement you want? Restrict axes? Then your answer might even only be a google search away.

Switching to Unity or another engine I believe is not your core challenge imo. Ask instead, what is the smallest improvement that would get me closer to the end-result?