Why hermitian, after all?

Hermitian operators (or more correctly in the infinite dimensional case, self-adjoint operators) are used not because measurements must use real numbers, but rather because we almost always decide to use real numbers.

As the OP mentions at one point, you might choose to use complex numbers to label a two-dimensional screen, and in that case you'll be able to use a so-called normal operator to represent the 2-dimensional observable. (Contrary to what Dirac thought, nothing whatsoever goes wrong here.)

It should not be too hard then to accept my next claim: You can use whatever measurement scale you want to measure a quantum observable! You can label pointer positions with items of fruit if you want to, and you can still build a perfectly legitimate observable.

There is no question that the reals and complexes have enormous advantages over other more arbitrary measurement scales (due to their rich internal structure which we capitalize on in the functional analysis), but the idea that real numbers are somehow endowed with a prestigious metaphysical status is baloney.

How to define an observable with any measurement scale you want to

Step 1. Set up a bunch of particle detectors

Step 2. Attach a label to each detector

The set of labels we'll denote by $\Omega$. Examples include: $$\Omega = \{0,1\},\mathbb{R},\mathbb{C},\{\heartsuit,\clubsuit,\diamondsuit,\spadesuit\}$$

Step 3. Write out the list of all possible events

By event, I mean a subset $\Delta$ of $\Omega$ that represents a possible question like "Did a detector in $\Delta$ fire?". We'll label the event structure $\Sigma$, e.g. $$\Sigma = \{\emptyset,\heartsuit,\clubsuit,\diamondsuit,\spadesuit,\heartsuit\clubsuit,\heartsuit\diamondsuit, \cdots,\heartsuit\clubsuit\diamondsuit\spadesuit\}$$

Step 4. Associate each event in $\Sigma$ with a projection operator

This is the hard bit, and there's no recipe for it. But you have to make sure that the family of projectors forms a Boolean algebra that perfectly mirrors the natural algebra of $\Sigma$.

We'll call the association $\sigma$, so that $\sigma:\Sigma\to\mathscr{P}(\mathscr{H})$.

And that's basically it! The object $\sigma$ (technically, a Projection Valued Measure on $\langle\Omega,\Sigma\rangle$) is a quantum observable. It contains all the probabilistic information you need to calculate the probability measure on your chosen measurement scale for any quantum state.

For example, suppose the state of the system is $\rho$ and you want the probability that a detector $\Delta\in\Sigma$ fires. The desired probability is just $p=tr[\rho\sigma(\Delta)]$.

What the heck does this have to do with Self-Adjoint operators?

Are you ready for the climax? Here it is...

IF you choose to use the measurement scale $\langle\mathbb{R},\mathscr{B}(\mathbb{R})\rangle$, THEN you will be able to build a self-adjoint operator which is precisely equivalent (in terms of the information it stores) to the PVM you constructed.

IF you choose a detection screen calibrated by $\langle\mathbb{C},\mathscr{B}(\mathbb{C})\rangle$, THEN repeat the above sentence replacing 'self-adjoint' with 'normal'.

(Neither of the above statements is obvious, by the way. They are famous results in Functional Analysis known as the Spectral Theorems.)

IF you choose to be a Fancy Nancy and use $\{\heartsuit,\clubsuit,\diamondsuit,\spadesuit\}$ for a measurement scale (with its power set for the event structure) THEN the fruits of your labors are more modest. In particular, you still get the answers to any questions you care to ask, but you don't get any neat operator to give you computational shortcuts. Instead you will forever be doing calculations like $p=tr[\rho\sigma(\heartsuit\clubsuit)]$.

I haven't even touched eigenvectors yet, but suffice it to say that they also do not have a fundamental status in the theory.

There is no doubt that we can learn something by reading the works of the great masters, but taking that work as the state of play can send you back a century. We've learned a lot since Einstein and Dirac.

When in doubt go back to the masters. From Dirac's Principles of QM

When we make an Observation we measure some dynamical variable. It is obvious physically that the result of such a measurement must always be a real number, so we should expect that any dynamical. variable that we can measure must be a real dynamical variable. One might think one could measure a complex dynamical variable by measuring separately its real and pure imaginary Parts. But this would involve two measurements or two observations, which would be all right in classical mechanics, but would not do in quantum mechanics, where two observations in general interfere with one another-it is not in general permissible to consider that two observations can be made exactly simultaneously, and if they are made in quick succession the first will usually disturb the state of the System and introduce an indeterminacy that will affect the second. We therefore have to restrict the dynamical variables that we can measure to be real..

According to Dirac property I in your question is more fundamental. Since Hermitian matrices are normal property II is satisfied. He also explains why a real dynamical variable should be a Hermitian matrix by starting from simple assumptions in his book. The book is quite old and is available for free.