seems to me that this should imply that a local observer standing on the earth (so not free falling at all) should be considered as an accelerating, non inertial frame.
Yes, an observer standing on the earth is not inertial in relativity. The definitive test is to have the observer carry a good accelerometer. In this case it will indicate an acceleration of 1 g upwards, conclusively showing that the observer is non-inertial.
Just a nitpick on language: an observer isnβt a reference frame, he or she has a reference frame, or even better there is a reference frame where he or she is at rest.
there is another, more geometrical, equivalent formulation of EEP:
Locally spacetime looks like π4
This is not the precise formulation of the geometrical formulation, but it's good enough.
Agreed, it is good enough for present purposes.
This means that in every sufficiently small region of spacetime it's like being into a inertial special relativity frame, so no accelerating, no gravity, no shenanigans.
It does not mean that at all. You can certainly have accelerating reference frames with pseudo-gravitational forces in π4. All π4 means is that you cannot have any tidal effects.
π4 is a flat spacetime manifold and can be equipped with an endless number of coordinate systems, including non-inertial ones. What βlocally spacetime looks like π4β means is that there exist local coordinates where the metric is the Minkowski metric (to first order), but it does not restrict you to using those coordinate systems.
More physically it means that tidal effects become negligible at small scales. The measurable effects from curvature, or tidal effects, are second order so they go away to first order at small enough scales.
But the geometrical formulation states that every sufficiently small reference frame, myself included, should be like an inertial SR frame!
No, the observer is unambiguously non-inertial. The geometrical formulation does not contradict that at all. The geometrical formulation merely says that in a small region spacetime is flat, not that an observer is inertial. It is perfectly consistent to have non-inertial observers and reference frames in flat spacetime. Only tidal effects are forbidden.
Best Answer
A derivation of Einstein's equation isn't why the Equivalence principle is central to GR. The reason that the equivalence principle is central to GR is in the fact that you can represent the gravitational field with a metric tensor at all--you can replace a force equation with a geodesic equation for a test mass precisely due to the fact that the geodesic that that test mass follows (or the "acceleration" felt by a Newtonian mass) is independent of the mass of that test$^{1}$ particle.
The equivalence principle, however, only selects out that one can represent gravity with a metric tensor. There are a great many other so-called "metric theories of gravity" that obey the equivalence principle, but are not general relativity--amongst other things, they will differ in the field equation for the metric tensor, or have extra fields in addition to the metric--the most famous of these is the Brans-Dicke theory, which treats Newton's constant as a scalar field coupled to the metric tensor. Most alternative metric theories have either been experimentally ruled out, or have had their additional fields constrained to the point where their values are consistent with zero (for instance, Brans-Dicke theory has a parameter $\omega$, and tends to GR if $\frac{1}{\omega}\rightarrow 0$. Current data says that $\omega > 4000$, or some similar number.).
$^{1}$Note that this is generally only true if the mass of the test particle is "small" compared to the local curvature of the spacetime, and if it's motion is slow enough to not produce gravitational radiation comparable to its energy. Either of these effects will cause the test mass to perturb the background spacetime, and those effects will both be mass dependent and cause the test mass to not follow a geodesic of the background spacetime. Both of these approximations are true (to great precision, at least) of all of the planets, asteroids and comets orbiting the sun, amongst many other things.