This was actually bugging me last night but my brain was too dead to quite be able to explain why... Doesn't help that I haven't taken a physics course in ages. It clicked for me today though.
I'm fairly sure the two points you indicate as the fulcrum and load aren't correct and that they're both fulcrum points working at the same time and the load is different areas of the head. The result of the two loads being pushed in opposite directions is that the head rotates in the direction of handle torque:
While you're correct that a longer eye puts more distance between the fulcrum and the load, this isn't the main reason it's advantageous. It's mostly that the higher surface contact is reinforcing the fulcrum points by spreading the load out over a greater area. The fulcrum points are after all where all the stress is condensed when force is applied. The greater amount of wood in the eye also just flat out means a stronger lever.
But all this is based around the idea that the handle is the lever and the head is the load. Realistically, the whole axe is one lever and the load we're trying to overcome is a stuck bit. When we yank up the handle, the head rotates around the fulcrum point in the bit. The toe of the bit is pushed against heavy resistance (whatever its stuck in) while the heel portion is freed (just like pulling a nail with the claw of a hammer):
It stands to reason that you'd also want to pull down on the handle to release the toe of the bit. Generally if its really stuck in there, you're going to be "pumping the handle" to progressively lower the resistance on either side of the fulcrum point in the bit until there is no resistance to be had and the bit is free. When you do this, the fulcrum points within the eye you mentioned alternate (green fulcrums when you're exerting effort in the white direction):
And then I come to the sort of embarrassing conclusion that I already knew. When faced with the
"stuck bit scenario" the answer is generally to
"pump the handle" - though now I can add the reasoning: instead of exerting all your effort in one direction and putting high stress on two fulcrums, you're exerting less force in alternating directions, wearing four fulcrums in lesser increments. Pretty much the equivalent of rotating your tires (wearing the eye evenly).