experimental physics - Can we measure an electromagnetic field?

As far as I can check, the Aharonov-Bohm effect is not -- contrary to what is claimed in the historical paper -- a demonstration that the vector potential $A$ has an intrinsic existence in quantum mechanics, neither that the magnetic field $B$ has an intrinsic existence in classical mechanics, but rather that the magnetic flux is the relevant, measurable quantity. Indeed, the phase difference is either $\int A \cdot d\ell$ or $\int B \cdot dS$ (in obvious notations with surface $dS$ and line $d\ell$ infinitesimal elements), and these two quantities are always equal, thanks to the theorem by Stokes.

Then, one could argue that classically one can measure the flux as defined above, whereas in QM one has only the Wilson loop $\exp \left[\mathbf{i} \int A \cdot d\ell / \Phi_{0}\right]$, with $\Phi_{0}$ the flux quantum, as a relevant, measurable quantity.

I just elaborated above about the Aharonov-Bohm effect since it was the first time I realised the possible experimental difference between measuring a flux and a field. Let me now ask my question.

My question is pretty simple I believe: Does anyone know a classical experiment measuring the magnetic field $B$ and not the flux $\int B \cdot dS$ ? As long as we use circuits the answer is obvious, since one can only have an access to the flux and the voltage drop $\int E \cdot d\ell$ ($E$ the electric field), but do you know other experiment(s)?

As an extra, I believe the same problem arises with force: *Does anyone know a classical experiment probing the force field $F$, and not the work $\int F \cdot d\ell$ ? *

NB: The question does not rely on how smart you can be to find a protocol that gives you an approximate local value of the field, say $\int F \left(x \right) \cdot d\ell \rightarrow F\left(x_{0} \right)$ with $x_{0}$ a (space-time) position. Instead, one wants to know whether or not a classical measurement can a priori gives a local value of the field, without a posteriori protocol. Say differently, one wants to know how to interpret a classical measure: is it (always ?) an integral or can it -- under specific conditions -- give a local value ?

Post-Scriptum: As an aside, I would like to know if there is some people already pointing this question out (wether it's the flux or the local magnetic field or vector potential that one can measure), and if it exists some reference about that in the literature.