Why does an airliner have a shallow descent when heavier, opposite to
gliders with ballast?
Actually, gliders share this characteristic with airliners, if we are talking about flight at some given high airspeed that is well above the airspeed for the best L/D ratio.
First a few general observations about gliding flight, based on something borrowed from another answer but now highly modified--
For simplicity, all the points below, and all the other content in this answer, ignore small changes in L/D ratio and glide ratio due to higher Reynolds number associated with higher airspeed. Likewise, effects related to Mach number or compressibility are also ignored.
For simplicity, all the points below, and all the other content in this answer, assume no airmass movement (no wind or lift or sink)
In this case, glide ratio is the same as L/D ratio
For a given aircraft in a given configuration, angle-of-attack determines L/D ratio. For a given aircraft in a given configuration, the best L/D ratio always occurs at one particular angle-of-attack. By "a given configuration" we mean a given shape, regardless of weight.
If angle-of-attack is held constant, increased weight increases airspeed, and increases sink rate, and decreases endurance (time to ground), but does not change L/D ratio or glide ratio or range (distance to ground). So the effect of an increase in weight is to shift every point on the airspeed-versus-sink rate graph toward a higher airspeed and a higher sink rate, moving each point along a straight line drawn from the origin of the graph through the point in question and then extended further onward. (The "GD" or "green dot" airspeed and sink rate are adjusted in exactly this way in figure "H4" included further below in this answer. The diagonal line represents the effect of increasing the weight while holding the angle-of-attack constant.) Therefore the glide ratio associated with any given angle-of-attack remains unchanged as weight is increased, but the airspeed and sink rate associated with any given angle-of-attack are both increased in proportion to the square root of the increase in the wing loading.
If airpeed is held constant, changing weight changes the L/D ratio and glide ratio, and also changes endurance (time to ground), and also changes range (distance to ground). At a fixed airspeed well above best L/D speed, more weight means a higher L/D ratio and a higher glide ratio and a lower sink rate and more endurance (time to ground) and more range (distance to ground), because the aircraft is being flown at an angle-of-attack that is closer to the angle-of-attack that yields the maximum L/D ratio.
Changing the weight changes the speed for best L/D ratio, but does not change the best L/D ratio that can be obtained.
Now on to the specific question --
The question cites some 737 performance figures for idle-power descent. In that case, the aircraft is flying an airspeed schedule that is dependent only on altitude, and is independent of aircraft weight, and is well above the airspeed for maximum L/D (i.e. for maximum glide ratio). (The exact meaning of ".78/280/250" is explained in this related answer.) It makes sense that the heavier plane will cover more ground than the lighter one, because the heavier plane is being flown at an angle-of-attack that is closer to the max L/D angle-of-attack than the lighter plane. Just as if you had decided that you were going to fly a speed run from 10,000' at 150 mph IAS in a glider, and the best L/D speed was 50 mph IAS in the unballasted condition and 70 mph IAS in the ballasted condition, you'd have a lower sink rate in the ballasted condition, and you'd also cover more distance over the ground in the ballasted condition, because you'd be flying closer to the max L/D angle-of-attack and airspeed.
If this seems counterintutive, see the second diagram in this related ASE answer. You can clearly see that at 190 kph, the heavier glider has a lower sink rate than the lighter glider. This is one of the fundamental reasons that gliders often carry ballast-- to give a lower sink rate when flying at some given, high airspeed, and thus optimize performance while racing.
This in no way implies that when flown at an airspeed selected to give the max L/D angle-of-attack, or any other fixed angle-of-attack, a heavier airliner experiences a different glide angle than a lighter one, or experiences a lower sink rate than a lighter one. Neither of those things would be true. When flown at a given angle-of-attack, such as the max L/D angle-of-attack, a heavier aircraft will achieve the same glide path as a lighter one, but the heavier aircraft will fly down that glide path with a higher airspeed and a higher sink rate.
One interesting thing about the 737 table is that the same time to descend from a given altitude is given for all weights. Perhaps this is close enough for a ballpark estimate, but in truth the heavier aircraft will take more time to descend from altitude, because the descent is being conducted not at a fixed angle-of-attack regardless of weight, but rather on a fixed altitude-versus-airspeed schedule that is well above best L/D speed. The only way the heavier aircraft can cover more distance while flying at the same altitude-versus-airspeed schedule as the lighter aircraft is to descend at a lower rate-- just like the ballasted glider flying at high speed in the examples above.
The question also cites a passage from an Airbus bulletin, including a reference to the "green dot" airspeed:
1.3.3. Weight Effect
Green dot speed (minimum gradient) is a function of weight. In the standard descent speed range (from green dot to VMO), the rate and
gradient of descent magnitudes are reduced at higher weights.
As explained in this related answer, the "green dot" airspeed increases with increasing weight, and serves as a good approximation to airspeed that will give the angle-of-attack for best L/D (best glide ratio). Therefore, upon first reading, the quoted passage may give the impression that the sentence "In the standard descent speed range (from green dot to VMO), the rate and gradient of descent magnitudes are reduced at higher weights" refers to the context of a descent conducted at some given target angle-of-attack, regardless of weight, referenced in some way to the "green dot" airspeed. However, this is not the case.
The first sentence on page 163 of the quoted Airbus bulletin begins "At a given TAS...". Furthermore, the first figure on page 163 of the quoted bulletin (figure "H4", reproduced below) suggests that the "green dot" airspeed does NOT play any role in defining the "standard descent speed range", and also shows that the "standard descent speed range" envelope is well above the best L/D airspeed regardless of weight, and also suggests that the "standard descent speed range" envelope is constant regardless of aircraft weight. Clearly, the phrase "the rate and gradient of descent magnitudes are reduced at higher weights" refers to a descent conducted according to an airspeed profile that is defined independently of weight, such as the ".78/280/250" profile discussed above. That is the reason that the descent rate, and the descent gradient, are reduced at higher weights-- in gliding flight or in low-powered descending flight at a given airspeed that is well above the max L/D airspeed, a heavier aircraft is being flown at a more efficient angle-of-attack than a lighter aircraft.
A final note: page 160 of the Airbus bulletin under discussion here states that "Descent is carried out at the Flight Idle thrust (i.e. at a thrust close to zero)." How do things change at higher power settings? At some higher power setting, it's clear that a heavy plane will be descending, while a lighter plane of the identical configuration flying at the same airspeed will still be maintaining altitude or even climbing. Therefore it logically follows that there must be some intermediate power setting where at some given airspeed, the exact same sink rate and glide angle are obtained at two different weights.
We've stated that "in gliding flight or in low-powered descending flight at a given airspeed that is well above the max L/D airspeed, a heavier aircraft is being flown at a more efficient angle-of-attack than a lighter aircraft". Unfortunately it never happens that in level powered flight, a heavier aircraft actually requires less thrust than a lighter aircraft of the identical configuration flying at the same airspeed. That would truly be "getting something for nothing"! In a sense, weight can be thought of as "fuel" (or more precisely, potential energy) for an aircraft descending at a low power setting or with zero power, but in no sense does weight act as a "fuel" for an aircraft flying horizontally (or climbing) with respect to the surrounding airmass.