I think you're missing air resistance there.

My hypothesis is that because the diaphragm gets thinner but its surface area remains the same, the effect of air resistance becomes more pronounced, and at lower frequencies, specifically, where the start/stop forces may not be enough to overcome this air resistance, the diaphragm won't move as far as it could.

Why would air resistance cause the diafragm to not have enough force ?
If that were the case we would see higher amounts of distortion as low frequencies would be flattened at the top at lower excursions already.
For e-stats flattening at low frequencies does happen simply because stretch of a tightly tensioned membrane is a force that 'holds back' excursions flattening sinewaves.

When the force on something is the same and the surface area is the same and the air resistance is the same the displacement is the same.
It just takes less extra energy to overcome the mass of thing pushing it.

So this "braking" happens due to natural air resistance, and not because of any electrical property. From an electrical standpoint, the conducting traces (and the cable) are the only things that should matter, so the amplifier won't necessarily see and be able to account for this mechanical incidence.

As said, the air resistance and mass remain the same for both drivers, as is the force, what would make the excursion differenent ?

If I have a spring that represents the air's resistance and use a thin, lightweigth (but rigid) material against it and apply 1N of force against it the spring will compress the same amount of percent as when the thin lightweight sheet was replaced by a thick and heavy plate of steel (when enough time is allowed)

Conversely, for e-stats, the capacitance will create a reactance at lower frequencies, which alters how much power is being delivered, but I don't think the amp is doing anything special to account for this air resistance.

No indeed the amp doesn't and that is my point. When you consider low frequencies close to static then you could see that if a membrane is sent to one side and held there it just stays there until something (electric charge or moving air) pushes it back. There is only a very weak static force that is strong enough to move the membrane and compress the air.
In case of current driven drivers the displacement is constant with a constant force. Completely different situation.

This can easily be remedied by increasing the "force" as frequency gets lower (do a bass boost), right?

Nope.

A hea dphone is more than just an electrical device after all.

That it is, it is a transducer after all. Electro Mechanical Acoustical with an immense amount of mechanical problems, more so than electrical ones.

I don't think thinner and lighter membranes will form a trade-off with LF response or distortion in linearity as a downside at all.
If anything it just extends the top end because the membrane has to 'consume' less power to be moved.
It simply can 'follow' the appllied current (or voltage) more accurately.

I think that by air resistance (minus the friction deal) Bill means pressure.

As far as diaphragm thinness and mass, the only thing I can think off is inefficiencies, practical considerations, and how much it can displace itself. If the driver has to pressurize a small enclosed volume, then the diaphragm proly does not have to move as much relative to free-field (specially if the surface area is relatively large). Maybe a thin diaphragm can't displace itself that much before deforming or going limp.

Regardless of this mass and HFM-nm-thinness talk, lower frequencies seem require more volume displacement than higher frequencies to maintain an specific SPL power. For a fixed area the only thing that can change is displacement, so that will increase. There is some talk about it here:

http://www.diyaudio.com/forums/multi-way/50769-spl-vs-frequency-vs-driver-movement.html

As far as what the amplifier can do, if the driver is physically unable to pressurize a cavity at a given frequency, my best guess is that all sorts of monkey chunks will fly and who knows what the amplifier will do... maybe self destruct in some cases.

Ports also change damping characteristics.

Following up on that... as Bill speculated: there's no free lunch. With awesome transient response, we must get limp-dick bass.

Lots of discussion here since I went back to sleep. I can't say I am getting all the Bill speak but I remember having a similar conversation some time back here

@OJ : I think what Marv is saying that the thin stat diaphragms when tensioned tend to gain an inertia which negates the speed/acceleration benefit of their lower mass. They end up requiring to overcome an initial threshold before they start moving which might mean that at some very low level signal the diaphragm may not move at all and end up losing that low level information.