Do observables only amount to computing functions of outcome probabilities? It is well known that in quantum mechanics any Hermitian operator $A$ can be thought of as an observable.
Given any (pure) state $\lvert\psi\rangle$, measuring such observable gives an average measured value of $\langle A\rangle_\psi\equiv \langle\psi\rvert A\lvert \psi\rangle$.
Being $A$ Hermitian, we can always write it as $A=\sum_k\lambda_k\lvert\lambda_k\rangle\!\langle\lambda_k\rvert$ with $\lambda_k\in\mathbb R$.
The average $\langle A\rangle_\psi$ can then be written as
$$\langle A\rangle_\psi=\sum_k \lambda_k p_k^A,\tag A$$
where $p_k^A\equiv\lvert\langle\lambda_k\rvert\psi\rangle\rvert^2$ are the probabilities of getting the $k$-th result when measuring $\lvert\psi\rangle$ in the eigenbasis of $A$.
My question is: do the eigenvalues $\lambda_k$ bear any direct physical meaning?
To be clear, with this I mean: would measuring a different observable $\tilde A$, differing from $A$ only in the eigenvalues, involve measuring the state in a different way? Like, in the real world, if you are to measure $A$ and then $\tilde A$ in your laboratory, do you have to perform different physical operations? Or is the difference instead only in the way the collected data is post-processed?
But then, if this is true, I wonder: why do standard treatments of quantum mechanics put so much emphasis on observables, rather then simply explaining that what is of direct physical significance is the probability associated with a state collapsing in a given basis, with the concept of observables just describing the act of post-processing in some way the collected probabilities?
This (unanswered) question also raises some points similar to the ones I am trying to make here (although it is more focused on the concept of POVMs).
Further discussions about this question on the hbar chat can also be found here.

Why the question?
The reason I ask is that in any situation I can think of, what one actually observes are samples resulting from having $\lvert\psi\rangle$ collapse in the eigenbasis of $A$. In other words, in the real world, one observes that the state collapses into one of several possible states (e.g., a photon's polarisation collapses to either $H$ or $V$ if appropriately measured).
Through repeated observations, we then end up measuring the probabilities $p_k^A$. To compute $\langle A\rangle_\psi$, we attach, somewhat arbitrarily, a number $\lambda_k$ to each of the outcomes. However, isn't this just a matter of useful convention?
In other words, doesn't this mean that the eigenvalues of a given observable are not actually a property of how the state is being measured, but rather of how we attach numbers to the different possible outcomes?
A practical example
Consider the act of measuring the polarisation of a photon. Say we are asking the photon whether its polarisation is horizontal or vertical (that is, we measure in the $\lvert H\rangle, \lvert V\rangle$ basis). I can describe this measurement with an observable $A$ that has eigenvalues $\pm1$ in the $\lvert H\rangle,\lvert V\rangle$ basis, that is, define $A=\lvert H\rangle\!\langle H\rvert-\lvert V\rangle\!\langle V\rvert$. How does measuring this work in the real world? You shine your photon, have it pass your polarisation filters or whatever else you want to use, and collect the associated statistics in your detectors. Are you ever using the values $\pm1$ in this process? I would argue you are not. What you do is collect statistics to recover the probabilities $p_H, p_V$, and then use these to compute $\langle A\rangle=p_H-p_V$. As you can see, the eigenvalues of the observable are using in post-processing of the collected data, but are not in any way a property of the measurement itself.
What I am not asking/questioning, and other common objections

*

*I am not by any means saying that the concept of observable is useless, or that it should be thrown away. It is undoubtedly a useful tool. I am instead asking whether it is strictly necessary, or whether instead one could get away in doing all the calculations without using it.

*What about the Hamiltonian? The Hamiltonian might be a special case, I agree. So let's say that the question refers to all observables except for the Hamiltonian.

*The eigenvalues clearly are the possible measurement results. This seems to be a common thought, but I would urge who thinks this to try and think of how any experimental measurement of a quantity actually proceeds. It is easier to understand in the case of measuring a discrete quantity such as the polarisation of a photon, but anything else would do as well. You do not measure "an eigenvalue" when you do an experiment. What you measure is one of the possible outcomes associated with the experiment. Each one of these outcomes is by you associated with an eigenstate/eigenvalue of some observable, but this is rather different than saying that one "measures an eigenvalue", as nowhere during an experiment the value of some $\lambda$ naturally emerges from the state. A similar point has also been made in the second paragraph of this answer.

*Think of measuring the position $\hat x$ of a particle: you obviously directly get the eigenvalues of $\hat x$ out of the experiment. I see why people say this, but the only difference with respect to the previous case is that this involves a much more ingrained convention. I would argue that even in this case the eigenvalues of $\hat x$ have nothing to do with how the measurement is performed. Think of a change of measurement unit, or a change of reference frame. This changes the eigenvalues of $\hat x$ but changes nothing in the underlying physics. Similarly, I can measure some "scrambled operator" $\hat x'\equiv\sum_x\pi(x)\lvert x\rangle\!\langle x\rvert$ (discretise the space at will to make this expression well-defined) with $\pi$ some permutation. How does the act of measuring $\hat x'$ differ from that of measuring $\hat x$? I would argue that there isn't any difference, if not in how, after the measurement data is collected, you choose to label the different experimental outcomes.

 A: *

*We don't measure only expectation values. In any particular measurement, we simply measure the outcome of that single measurement, and, that measured value can only be one of the eigenvalues of the observable you are measuring. So, yes, the eigenvalues of a Hermitian operator have a direct physical meaning, they are the values that you get when you measure the corresponding observable. 

*We don't attach the numbers $\lambda_k$ arbitrarily with the probabilities $p_k^A$. $\lambda_k$s are the eigenvalues of the Hermitian operator $A$ and you simply cannot choose the eigenvalues of an operator arbitrarily. More physically, when you measure, say, the momentum of a particle, the fact is that it will turn out to be some specific number, one of the eigenvalues of the momentum operator. For a particle on a ring, it will have a definite spectrum and for a particle on a line, it will have another spectrum. You can't choose them arbitrarily.

*Moreover, via measuring 
different observables for a given state and determining different probabilities $p_k^A,p_m^B,..$ we don't gain any information about the operator itself. We gain information about the state. The operator (and its spectrum) is supposed to be already known. 

