Quite simply you are right. Tidal forces are not subject to the equivalence principle.
If you are an accelerating reference frame, then you can represent space with the Rindler coordinates. The bottom line is that, no, of course this can't possibly be equivalent to the field you see around a body like Earth. It just can't even represent the same universe.
Rindler coordinates contain the parabolic motion (actually hyperbolic) motion you're familiar with in a constant gravitational field. But here's the more interesting thing about the Wikipedia article I linked to, pay close attention to this wording:
the Rindler coordinate chart is an important and useful coordinate chart representing part of flat spacetime, also called the Minkowski vacuum.
Now, there's something called Minkowski spacetime. It's GR-talk for "flat space". Notice that Rindler coordinates don't replace the Minkowski spacetime. Rindler coordinates still are Minkowski spacetime. A transformation (made possible by the equivalence principle) gets you from one to the next.
The question refers to gravity due to a spherically symmetric object. This "metric" (like a map of spacetime) that corresponds to this is the Schwarzschild metric. This metric is fundamentally different from the (flat) Minkowski spacetime. That's not true going from Rindler to Minkowski, they are the same.
The equivalence principle allows a coordinate transformation between reference frames that do or don't claim to have a gravitational field at their point in space. So there's no universal answer to "what is the gravitational field here?" However, there is a certain kind of invariant topology to spacetime that absolutely doesn't change between reference frames. This topology basically corresponds to tidal forces. Tidal forces gives rise to spacetime curvature, but a field alone does not.
I think of it like this: gravitational field is like the "slope" of space. You can change your body direction and see a different slope! However, tidal forces are like the "curve" or space. It doesn't matter which way you're pointed, you must agree on the presence of curvature.