Some of the late Victorian households in the UK were great fans of brush coating furniture with dark shellac. This method of furniture care was used as a way of easily refreshing the furniture. Maybe wax polishing was too much effort for them? After all, in many cases the furniture would be decades old and only really serve them as functional, utilitarian pieces. So to refresh the old (French Polished) finish, a 'refreshing' brush coat of a weak mixture, coloured shellac will do the trick nicely.
It would have been a weak solution brushed on and left to dry, which would have done so quite quickly. I would imagine the shellac was stored in a jar, as we do now, but without much worry whether it had dust and dirt in it.
So what happens a generation later is this finish starts to break down and flake away from the surface, sometimes to reveal the original finish underneath and sometimes it will flake away, taking the primary finish with it leaving the exposed original wood.
At this point the aim was to prevent the finish from flaking further so a coating of 'something' was applied to the chair finish to bind it all together and prevent further flaking. The 'something' could have been a varnish or shellac, I suspect it was whatever was at hand.
From the look of the images this is what has happened to these (museum) chairs. This is why when you are 'spot testing' them it's not penetrating any deeper. It's almost impossible to prevent the flaking by applying anything to the surface.
The finish in images looks like it hasn't had this later application of 'binder' and the finish is flaking away.
The most sympathetic course of action would be applying our antique gold beeswax polish to them. Because it is so rich in its content it will help capture/retain the old finish. Be careful to protect the upholstery.
I think it's your only option if the aim is to preserve this finish. Applying a film coating like shellac or an oil based coating will mean you are effectively altering the finish. but wax polishing is not a film coating, it's a sacrificial enhancing layer, that over time will wear away, and this will take years if they are not used. It's primarily acting as a conservation wax.
The tiny amount of stain in the polish will help tone down the white exposed areas, but only a little.