Thank you, glad you enjoyed my game!
You definitely want to find a way to limit player movement, my first idea was actually very similar to yours; having a platform you can drag around to use, then I realized it would be a little too broken and much harder to make thought provoking puzzles. Also If you are going to use UI as a platform in your game I would recommend trying to find a way to emphasis that it is actually UI rather than just a platform you can drag around. For me I included the camera zooming mechanic which changed the size and position of UI platforms relative to the environment and then made puzzles based around that idea.
Also for your game I see no reason to have a separate UI mode to be able to move the platforms. Switching back and forth felt tedious considering that I could still just move the platform around while the player was moving. And I still couldn't figure out how to get the screen size right. The edges of the screen are cut off which cuts off part of the pause button. I also use unity and i think you just need to adjust the resolution of the game in the project settings to match whatever resolution you are playtesting in.