I wonder if a suspension seat post of this style can potentially extend the life of a bicycle frame that often get off road and into gravel ground and sometimes submitted to going up and down the curbs.
Depends on what you compare to.
If you compare to a skilled rider who is not tired, the skilled rider would anyway use leg muscles to dampen all shocks, if riding on uneven ground. Compared to such a skilled rider, there is no extra protection.
However, if you compare to a tired and/or careless rider, who is just sitting on the saddle even though the ground is uneven, then the suspension seat post is a massive benefit to the fatigue life of the frame.
So yes, a suspension seat post can increase the lifetime of a frame. The way frames fail is by fatigue (exception: carbon fiber doesn't fatigue), and by reducing the forces the frame sees increases the fatigue life, perhaps in the case of steel even from a finite number of kilometers to an infinite number of kilometers. Aluminum always has a finite fatigue life, but the increase in frame lifetime can even be 100x - 1000000x (or even more!) 1 if you use a suspension seat post.
Carbon fiber frames fail by encountering forces they were not designed to withstand. In this case, I suspect a suspension seat post could be of benefit too, since there's always a risk that if riding too fast on uneven ground on a crazy-light carbon fiber frame, that the forces the frame would need to encounter exceed its design limits.
(1): Yes, this figure is correct. See the aluminum fatigue life curve here: https://en.wikipedia.org/wiki/Fatigue_limit
Note that decrease from 45 ksi to 25 ksi increases number of cycles 100-fold, and that decrease from 45 ksi to 15 ksi increases number of cycles 100000-fold. It is perfectly reasonable to assume that a suspension seatpost would reduce the loads the frame has to withstand to a value even much smaller than one third of the loads in a non-suspension-seatpost-frame. Of course, this applies only to fatigue life. For example if you increase the fatigue life of your frame million-fold, but at the same time don't ride more carefully nearby cars, it is possible you only experience 10x the frame life since a collision with a car kills the frame. Also, on aluminum frames you will eventually have worn away bottom bracket threads, which is the end of the frame, and the bottom bracket threads don't care that your frame has million times the fatigue life due to a suspension seatpost, they will wear out at their natural rate anyway.
So the figure 100x - 1000000x means for fatigue life. There may be other reasons a frame fails.