Exporting to ZIP/PCK gives you a package of all the data files for your game, but not the Godot engine itself. If you gave someone the ZIP/PCK, they would have to go and download Godot themselves to run it. (If you've ever played old Doom levels, the ZIP/PCK is like a WAD file, while Godot is the program itself.)
The EXE is so large because it's got the whole Godot engine in it. You can make it slightly smaller by toggling off the option to include debugging when you export it, but it doesn't change the size by much. I don't think compiling Godot yourself would change the size of the exported EXE, as that's how it's coded.
Unfortunately I don't think you can make a game in Godot that's smaller than 10 MB.