Cool game, congrats! The leaderboard is an amazing feature because it really encourages risking as much as possible. The level generation adds a lot of replayability, although something like a wall jump would have probably been more important for a game jam entry. I tend to spend a lot of time on not entirely needed level generation too lol, I totally get it, it's a lot of fun to program.
As for the stutters, I am almost certain this is shader compilation, although it is strange that it happens again when restarting, it should have been still cached as far as I know. The initial stutter is compilation of the debris particle material, and then the particle material of the coin collection effect.
If I am right, the same fix I did in our game (also in Godot) will work: You add a tiny SubViewport set to render once and never clear and add a Sprite2D with each shader and a GPUParticles2D with each particle effect as children of that viewport. They need some texture, I just use a white 1x1 pixel. This will render them to that viewport, never to be seen by the player, but it forces compilation of these materials at the start of the game, so it doesn't need to happen at runtime.