In what sense (if any) is Action a physical observable? Is there any sense in which we can consider Action a physical observable?  What would experiments measuring it even look like?  I am interested in answers both in classical and quantum mechanics.
I ran across a physics textbook called "Motion Mountain" today, with volumes covering a broad swath of physics, written over the past decade by a dedicated German physicist with support from some foundations for physics outreach. So it appears to be a serious endeavor, but it's approach to many things is non-standard and often just sounds wrong to me. In discussing his approach to some topics the author says:

On action as an observable
Numerous physicists finish their university studies without knowing that action is a physical observable. Students need to learn this. Action is the integral of the Lagrangian over time. It is a physical observable: action measures how much is happening in a system over a lapse of time. If you falsely believe that action is not an observable, explore the issue and convince yourself - especially if you give lectures.

http://www.motionmountain.net/onteaching.html
Further on the author also discusses measurements of this physical observable, saying

No single experiment yields [...] action values smaller than hbar

So I think he does mean it literally that action is physically measurable and is furthermore quantized. But in what sense, if any, can we discuss action as an observable?

My current thoughts:
Not useful in answering the question in general, but hopefully explains where my confusion is coming from.
I will admit, as chastised in the quote, I did not learn this in university and actually it just sounds wrong to me. The textbook covers classical and quantum mechanics, and I really don't see how this idea fits in either.
Classical physics
The system evolves in a clear path, so I guess we could try to measure all the terms in the Lagrangian and integrate them along the path. However, multiple Lagrangians can describe the same evolution classically. The trivial example being scaling by a constant. Or consider the Lagrangian from electrodynamics which includes a term proportional to the vector potential, which is itself not directly measurable. So if Action was actually a "physical observable", one could determine the "correct" Lagrangian, which to me sounds like nonsense.  Maybe I'm reading into the phrasing too much, but I cannot figure out how to interpret it in a manner that is actually both useful and correct.
Quantum mechanics
At least here, the constant scaling issue from classical physics goes away. However the way to use the Lagrangian in quantum mechanics is to sum over all the paths. Furthermore, the issue of the vector-potential remains. So I don't see how one could claim there is a definite action, let alone a measurable one.  Alternatively we could approach this by asking if the Action can be viewed as a self-adjoint operator on Hilbert space ... but the Action is a functional of a specific path, it is not an operator that acts on a state in Hilbert space and gives you a new state. So at first blush it doesn't even appear to be in the same class of mathematical objects as observables.
Ultimately the comments that experiments have measured action and show it is quantized, make it sound like this is just routine and basic stuff I should have already learned. In what sense, if any, can we discuss action as a physical observable? What would experiments measuring it even look like?
 A: The book propagates a myth.
Experiments measure angular momentum, not action - even though these have the same units. One finds empirically that angular momentum in any particular (unit length) direction appears in multiples of $\hbar/2$, due to the fact that its components generate the compact Lie group SO(3), or its double cover U(2).
That Planck's constant $\hbar$ is called the ''quantum of action'' is solely due to historical reasons. It does not imply that the action is quantized or that its minimal value is $\hbar$. Early quantum theories such as Bohr-Sommerfeld approximation used quantized action but this was an approximation to the more general quantization of Dirac, etc... One must additionally take care not to confuse the word action in "action-angle coordinates" with the action of variational calculus.
Indeed, the action of a system defined by a Lagrangian is a well-defined observable only in the very general and abstract sense of quantum mechanics, where every self-adjoint operator on a Hilbert space is called an observable, regardless of whether or not we have a way to measure it. The action of a system along a fixed dynamically-allowed path depends on an assumed initial time and final time, and it goes to zero as these times approach each other - this holds even when it is an operator. Hence its eigenvalues are continuous in time and must go to zero when the time interval tends to zero. This is incompatible with a spectrum consisting of integral or half integral multiples of $\hbar$.
A: In the language of the OP, action is a functional, since it is an integral of the Lagrangian... but over an arbitrary path. In other words, it is an abstract mathematical object, which has no counterpart in the real world.
This functional is then minimized in respect to all possible trajectories. In quantum mechanical terms the action along the optimal trajectory corresponds to the phase of a wave function, which is measurable (although defined up to a constant), e.g., in the experiments on Aharonov-Bohm effect, and any other interference experimente. The fact was recognized long before the advent of the path integrals - Landau&Livshitz derive quasiclassical approximation is an eiconal expansion of the phase of the wave function, which they openly call action.
A: One can measure an on-shell action by
counting cycles and noting the phase within the cycle.
Details:
$∂τ$ is a proper time step of the system
consisting of observable simultaneous components.
By observing the components one can find repeated patterns.
Basically all systems cycle.
$τ=∫∂τ$ is the constant information of the system or
Hamilton's principal function.
$$0=\frac{dτ}{dt}=\frac{∂τ}{∂x}\;\frac{∂x}{∂t}+\frac{∂τ}{dt}\\
W=\frac{∂τ}{dt}=-\frac{∂τ}{∂x}\;\frac{∂x}{∂t}=-H\\
H=pẋ=mẋ²$$
Note,
$p = mẋ$ is by observation, i.e. the physical content.
Then it is assigned to $\frac{∂τ}{∂x}$ by definition,
leading to $H=mẋ²$.
$∂τ=mẋ∂x$ then says
that $ẋ$ and $∂x$ contribute independently to a time step $∂τ$.
Splitting off some non-observable part of the system
and associating it to the location of the observed part
$H(ẋ,x)=T(ẋ)+V(x)$ half-half, makes $T(ẋ)=mẋ²/2$.
Half-half is the usual but not mandatory choice for $V$.
$W=-H$ stays constant.
It is the cycle's information divided by the period time.
$I=∮(∂τ/∂t)dt=∮(-H)t=Wt$.
One cannot minimize $∫Wdt$ because it increases monotonously,
counting until the system ceases to exist.
$L(x)=mẋ²+W=mẋ-H$ oscillates and returns to 0 in a cycle.
$J=∫Ldt$ returns to the same value after one or many cycles.
Minimizing this produces conditions
to attribute observables to the same system time step
(the equations of motion).
$$0 = \frac{δJ}{δx} = \frac{1}{δx}∫\left(δx\frac{∂L}{∂x}+δẋ\frac{∂L}{∂ẋ}\right)dt = \frac{1}{δx}∫δx\left(\frac{∂L}{∂x}-\frac{d}{dt}\;\frac{∂L}{∂ẋ}\right)dt \\
\frac{∂L}{∂x} = \frac{d}{dt}\;\frac{∂L}{∂ẋ} \\
F=ṗ$$
One can measure an action satisfying the equations of motion
by counting cycles and noting the phase within the cycle.
Measuring our every-day time is also via measuring action.
Comparing system changes to our time unit motivates energy $W$.
$W=\frac{∂τ}{dt}$ is system time divided by our time,
which is frequency times a factor to keep the units consistent.
