Here's my position.
To maintain flight the aeroplane must generate lift. To generate lift it incurs drag and thus must provide thrust.
A heavier plane must increase its angle of attack to provide extra lift. It thus incurs proportionately more drag and thus requires proportionately more thrust.
The thrust comes from the vertical component of the glide ie the rate of change of height times the weight of the plane.
So the heavier plane needs more thrust because it is heavier.
But it is heavier so it provides more thrust for a given angle of descent.
What I am saying here is you don't have to 'pay' more to glide a heavy plane. You have already paid for the weight when you climbed.
This leaves you at the mercy of the wing designer as to whether you will glide better or worse at higher wing loads. Your 737 may do better heavy. Another plane might do better light.
