Firstly, let's get some terminology straightened out...
There are 3 distinct connections for mains. They are (1) Active (or Live), (2) Neutral and (3) Earth (or Ground).
The Active and Neutral are connected directly to the mains supply and both should be treated as potentially lethal. In actual fact, the neutral should be connected to an earthing point somewhere near the distribution transformer - which means there should not be any significant voltage on it when measured with respect to a local earth. You should NEVER rely on this for any matters of safety, since there are several conditions that can result in voltage on the neutral.
The Earth connection is completely separate. It is a connection to an earthing rod which has been hammered into the ground somewhere near your meter box. In years gone by, they used to use a water pipe - but that caused an electrocution risk to plumbers.
Do NOT ever rely on the Earth and Neutral as being identical - especially in regards to safety.
To answer your original question: The element will be isolated from any external metalwork so it doesn't really matter which way the plug is connected. That alligator clip will be acting as a local earth to the board minimising any risk from static electricity and induced currents - but that is all. Do not ever think that this thin cable or clip are capable of acting as a safety earth. Even if attached to a proper, adequate earthing point (which is often not the case), they are far too flimsy.
While rewiring the unit to have a 3 pin mains plug and proper earthing could be worthwhile, there is not a lot of room inside these things for such engineering changes and I would not feel comfortable advising someone to try it unless they have some experience with mains. Since I get the feeling you don't have such experience, my suggestion is to not try it.