With Jet transports, a light aircraft falls out of the sky - with the 747 we used an adjustment (based on weight) to determine our top of descent.
Only because, in those circumstances, it's not desirable to reduce the speed by the amount needed to maintain glide range.
To justify it aerodynamically, you need to start with a graph of L/D versus Lift Coefficient - any good text book will provide. This shows L/D rising as CL increases until it reaches a peak, then falling for further increases in CL.
CL is proportional to W/(V^2), so (L/D)max occurs at a given W/(V^2).
For reduced weight, the glide range will either a) reduce if the speed was high enough to put you to the left of (L/D) max (which would be the case for a cruise descent), or b) increase if the aeroplane was flying slow enough to be on the right of (L/D)max at the reduced weight.
For GA single-engine flying, the most important case is achieving maximum range following engine failure. Here, you want to glide at the speed that gives (L/D)max ... and the glide range (nil wind) will then always be the same, regardless of weight. So to achieve the W/(V^2) that gives (L/D)max, if W is reduced then V needs to be reduced (although only by 0.5-0.6x the weight reduction %, because V is squared).
As an example, in my Bonanza a glide at (L/D)max gives a nil wind glide range of 1.7nm per 1000ft. The glide speed to achieve that, however, is 15kts lower when the aeroplane is at its lightest compared with when it's at MAUW - circa 95kts versus 110kts.