I did a mini-review of the Berthoud Aspin. Is that the model you are concerned with, Sorefeet?
When you sit on that seat, those rails aren't going anywhere. Your weight on the seat will compress the nose and crescent onto the rails. Any residual tensile stress in the leather will ensure the rails stay put when you're not seated. I think your test, removing the rails while off the bike, is an unrealistic condition, relative to the functionality as a saddle.
I think your test is probably a better representation of a desireable feature from a manufacturing perspective rather how it's it's to be ridden.
The leather does, indeed, look to be very thick. Is yours as thick? Look at the right side of this photo, where the leather meets th ecrescent, to appreciate the thickness of the leather.
This also shows how thick it is.
It does not appear to be a leather cover stretched over a plastic saddle. It appears to be a leather saddle stretched between two plastic pieces - a nose and a crescent. As such, I'd say it is a leather saddle.
No matter the thickness of the leather, what determines how stiff the seating area is is the modulus of the leather. If the leather is low modulus, you have a soft seat. If it is high modulus, you have a stiff seat. You can adjust the stiffness by adding preload to the leather but this has limits. If the leather is too soft, it will just stretch and stretch as you add tension. If it's too hard, you will likely need to reduce preload.
One thing I learned about leather (learned the hard way) is NEVER use oil on the leather, especially Neat's Foot Oil. Oils destroy the leather in very short ordder. If the leather looks dry, it is. This means it will soak up oil like a sponge! At that point, you can kiss the leather goodbye.
P.S. - I just looked up Obenhauf's. It is an oil not saddle soap, not lanolin, not Proofhide, not even the material Berthoud provides for their saddles.
P.P.S. - Ah!, Obenhauf's also makes a paste. Which did you use?