You're right that this is glossed over in popular explanations, but the point you make is exactly the starting point for all formal courses and textbooks.
The first part of the argument is that for single point particles falling, the effect of gravity is the same for all particles. This suggests that we should model gravity as something intrinsic to spacetime itself, rather than as a field living on top of spacetime, which could couple to different particles with different strengths.
The second part of the argument, which is what you point out, is that gravity can have nontrivial tidal effects. (This had better be true, because if all gravitational effects were just equivalent to a trivial uniform acceleration, then it would be so boring that we wouldn't need a theory of gravity at all!) This suggests that whatever property of spacetime we use to model gravity, it should reduce in the Newtonian limit to something that looks like a tidal effect, i.e. a gradient of the Newtonian gravitational field. That leads directly to the idea of describing gravity as the curvature of spacetime.
So both parts of the argument give important information (both historically and pedagogically). Both parts are typically presented in good courses, but only the first half makes it to the popular explanations, probably out of simplification.
"Tidal effects" are defined in terms of having different gravitational fields in one place than another (i.e. the tidal bulge near to the moon occurs because the moon's field is stronger there).
The first part of the argument is that for single point particles falling, the effect of gravity is the same for all particles. This suggests that we should model gravity as something intrinsic to spacetime itself, rather than as a field living on top of spacetime, which could couple to different particles with different strengths.
The second part of the argument, which is what you point out, is that gravity can have nontrivial tidal effects. (This had better be true, because if all gravitational effects were just equivalent to a trivial uniform acceleration, then it would be so boring that we wouldn't need a theory of gravity at all!) This suggests that whatever property of spacetime we use to model gravity, it should reduce in the Newtonian limit to something that looks like a tidal effect, i.e. a gradient of the Newtonian gravitational field. That leads directly to the idea of describing gravity as the curvature of spacetime.
So both parts of the argument give important information (both historically and pedagogically). Both parts are typically presented in good courses, but only the first half makes it to the popular explanations, probably out of simplification.