What is the quantifiable benefit of compressionless brake housing?

In a static situation, and with the same force applied to the lever, in all likelihood the ultimate braking force will be the same between compressionless and regular housing. But that's only part of the story.

Housing compression essentially adds an element of "spring" or slack to the system. This extra amount of spring could cause a few different effects.

• All other things being equal, extra spring in the system means the lever has to travel further to apply the same amount of brakes. This could result in the lever running out of travel sooner, or moving into a range that is less efficient for your fingers. You could tighten the cable a bit to mitigate that, but it will then cause the brake pads to rub sooner. Since the lever has to move further to overcome the housing compression, it also means the cable moves further through the housing which should introduce more friction and wear.
• Extra spring in the system slows down haptic feedback to the human. This makes the brakes feel less precise. When brakes are less precise, there is a human tendancy to be more cautious with them. Very precise and repeatable brakes tend to result in humans applying more braking power, and applying it sooner, even if the ultimate braking power is the same. This has been shown also with cars; we know that sometimes accidents happen even when the car is mechanically capable of stopping in time, but the driver didn't apply the brakes hard enough or soon enough. Some cars are being equipped with automatic braking features for this reason.
• The extra spring from housing compression is not a perfect constant and it can change with cable movement. This is one of the reasons linear housing was invented for shifting, to prevent the shifting point changing either over time or as the handlebars were turned or cable routing flexed. The same thing that applies to the shifting point applies also to the braking point. So we can't really model housing compression as a spring. It's more like a spring that changes randomly and includes varying amounts of friction to go with it.

Basically, none of the effects of extra spring/slack in the braking system cause improvements, so it's better to get rid of it.

About the "linearity" of housing compression, experience suggests that once the housing compresses initially, then it is somewhat more resistant to further compression, so it's not like a spring that is subject to infinite, linear compression. It also means that if housing could be theoretically "pre-compressed" and maintained that way somehow, that it could improve braking performance and might be an alternative to linear housing. Also, it's possible that different brands of housing might have a more precise winding and therefore have a slightly better compression trend. I'm not aware of any measurements to quantify the compression curve of brake housing, but that would be interesting to see. It could even be specified roughly as a certain amount of compression per meter per Newton.

Linear housing greatly reduces the effects of housing compression and generally makes cable brakes better as a result, but do note that linear housing is much less flexible, and it can cause problems with kinking at the barrel adjusters or causing self-steering by causing a force pushing handlebars off-center. So it's best used on bikes with gradual cable routing, and won't work where tight bends are needed.

Even with compressionless housing the best housing should still be kept the shortest possible. Even compression-free housing unless it consists of non-compressible metal beads will still have residual compression. This will carry as much as possible of the force applied to the cable to the callipers.

From the front-lever to the brake, there isn't obviously any further shortening possible, unless you don't want to hamper the correct functioning of the steering.

With the rear brake you go from the lever to the point where the housing sits in a stop on the top tube, let's say, then runs free next to the seatpost where another cable stop picks up a shortest possible length of housing to join the rear brake (this may of course be externally or internally in the tube). Compare this to the old-time method of running the housing all the length of the top tube. Even with compressionfree housing, the old way will leave some sponginess in the rear brake, removing efficiency and feel.

There are two types of mechanical brakes: short pull and long pull. Short pull brakes include cantilevers and caliper brakes, plus some mechanical disc brakes. Long pull brakes include V brakes plus some mechanical disc brakes.

Short pull has less cable pulled, but the force in the cable is larger so the product of force and cable travel is the same for both pull types.

The type of outer cable traditionally used for mechanical brakes is coiled steel outer cable. It withstands the large braking forces. The outer cable types used for shifting are sometimes called compressionless housing, but this is an incorrect term. Instead, they should be called constant-length housing because the length of the housing does not change if the bending radius of the housing changes. This property is important for shifting because it is what allows you to use handlebar mounted shifters. Otherwise the turning of handlebars would affect the gear you're using.

Good manufacturers also divide the coiled steel housing class for different brake types: short-pull brakes and long-pull brakes. For example, Shimano has SLR housing for short-pull brakes and M-system housing for long-pull brakes.

The short-pull housing has been optimized to be dimensionally stable even if the forces transmitted by the cables are large.

The long-pull housing has been optimized to cause as little cable friction as possible. This is important, because the mechanical advantage of long-pull brake levers is such that it would cause cable friction to deteriorate braking feel.

You probably can substitute short-pull housing for long-pull brakes but the braking feel can be less than optimal. Substituting long-pull housing for short-pull brakes probably works for lightweight riders or riders not having strong fingers. However, a strong-fingered heavy rider could exceed the design forces of long-pull housing if used with short-pull brakes. So that's why riders needing large braking forces shouldn't use long-pull housing on short-pull brakes.

I don't understand why anyone would like to use constant-length ("compressionless") housing on any brake.

It's about distance, not force. Coil housing is not at all impaired in its ability to transmit force once any distance between its coils has been taken up (so no to your second question.) On any rim brake that doesn't have egregious bends, the practical difference between compressionless and coil housing is marginal. It's a little stiffer and more responsive feeling, but the advantages of it are small and possibly outweighed by the increased fragility, installation hassle, and/or expense of compressionless. On a mechanical disc, or a rim brake system that forces egregious bends in the housing, it's a different story.

Total pad gap on a V-brake or dual pivot caliper is around 5mm. On a mechanical disc it's 0.8-1mm. Yet the lever throw powering both is the same; the needs of a brake lever designed for a human hand doesn't change between the two. This demonstrates that in terms of lever movement to pad movement, mechanical disc brakes run at about a 5 times greater leverage ratio, or mechanical advantage. You have a great big lever throw producing a very small amount of movement. That's how disc brakes are powerful despite a rotor being such a smaller circle than a rim. Mechanical discs can have problems with being difficult to set up in a way where the rider doesn't run out of lever travel, and since compressionless helps conserve lever movement by eliminating housing compression, it is good to use in this role and is often one of the main ways of helping out a bike that is having issues with the levers bottoming. This is often most pronounced on long continuous rear brake housing runs, which most disc bikes have.

The effect of coil housing creating wasted lever movement becomes much more pronounced whenever there are tight bends in the housing path, which will cause the gaps between the coils to open up and need to be closed again by the braking force. The helical nature of the individual strand path in braided compressionless brake housing mitigates this effect almost entirely in most cases on bikes. The same effect can sometimes be used to create good brake feel on bikes with brake routing that forces awkward bends.

It is worth pointing out that compressionless brake housing does have a dark side compared to coil. Coil housing, being made of thick steel, is a very robust design that is much more able to shake off or only be incrementally affected by damage, abrasion, kinks, and material degradation. Compressionless brake housing is something of a glass cannon in comparison; it's a bunch of steel wires that fundamentally don't want to stay together and resist the forces they're being put under, but are able to do so because of the standout properties of the layer of kevlar fabric that constrains them. The system is also wholly reliant on the right special ferrules being used to keep wires from pulling through. Compressionless brake housing is safe, more than reliable enough for bikes that are reasonably maintained, and not prone to failing suddenly when it does fail, but it is less age- and abuse-resistant than coil.

EDIT: As ojs pointed out in the comments, I had the part about the effect of compression being amplified 5x wrong, which I initially wrote. I've edited this answer to be something I can stand by, but see that it was likely accepted on the basis of incorrect reasoning, so I'm open to it being taken down or un-accepted as the answer, neither of which I can do.