I downloaded and tested the project with no noticeable lag on Godot 3.0.6 on Windows 10. When I am running the main scene (not the menu), I get 2869 frames according to the debugger.
For reference, here is my system specifications: Intel i7-3770 3.4 GHZ, 8 GB Ram, Nvidia 660 TI 2gb.
I tried to run the project in the profiler, but my computer was generating too many frames and the profiler could not keep up. I had to turn on VSync and then I was able to profile the project. Looking at the profiler, I did not see anything that would cause any lag. The profiler showed a pretty stable line with nothing causing any spikes to performance.
The only time I saw anything strange was when the player is jumping. When jumping, the player kind of snaps from position to position. At first I thought this was lag, since it kinda looked like it, but the profiler doesn't show any spikes in performance or FPS drops.
So I decided to take a look at the code, and I think I found the line of code responsible for the lag like behavior when jumping in the player script:
onAirTime += 0.01666 # 60 frames per second, 1 = 1 second
I changed it to the following:
onAirTime += delta
And then jumping was smooth and like you would expect. Not sure if that was the problem, but other than the jumping thing, I did not notice anything else.