There's a difference between accelerating compared to a coordinate system, and accelerating compared to an inertial rest frame. If you set a fixed coordinate system relative to the ground, then yes, the cat is accelerating towards the ground.
Consider this: if you hold an accelerometer while stationary on the ground, it will read 1g (accelerating). If you read an accelerometer while in free-fall, it will read 0g (ignoring wind resistance). In the first scenario, you are accelerating compared to your rest frame, even if you are standing still.
This is not an arbitrary distinction either. The 1g of acceleration while stationary on the ground produces measurable relativistic effects.
An accelerometer doesn't measure what you think. It measures the difference between the acceleration of the casing and the internals, for an example. It is designed to show 0 in free-fall. Yet, it still is under the force of 1G towards the center of the Earth.
There's more nuance to it than that, so let me try to clarify again. Like you said, the accelerometer shows the difference between the casing's reference frame and the internal's reference frame. The internals ideally maintain an inertial reference frame. It's only "designed" to show 0g in free-fall because the difference between the reference frames of the casing and internals _is_ 0 when you are falling. Because, again, when you are falling, you are in an inertial reference frame - you are not accelerating against the flow of spacetime.
The only time the accelerometer reads something other than 0 is when something pushes it away from an inertial reference frame. This is true when you're on the ground: spacetime is curved, and "flows" towards the center of the Earth. The ground pushes you against the flow of spacetime, and the difference between these two frames of reference is 1g. Note that this 1g is not "caused" by gravity, it's caused by the electromagnetic force. Forces push matter away from an inertial reference frame. Gravity is different. It decides where an inertial reference frame goes by curving spacetime - it is not a force.
Hypothetically, if you were are the center of the Earth, the accelerometer would read 0g. There would be nothing pushing you in any particular direction - you would be in an inertial reference frame (ignoring the minor detail of being crushed by all of the Earth's mass).
Again, I'm trying to show the subtle difference between accelerating compared to a fixed coordinate system, and accelerating compared to an inertial reference frame. The fixed coordinate system does not take into account the curvature of spacetime. If it did, the coordinates would be "accelerating" towards the center of the Earth at 1g - congratulations, you've defined an inertial reference frame.
I hope that makes sense. It blew my mind when this idea clicked: that space and time are not two distinct things, they are two parts of the same thing.
Well, yes, but I never disputed any of that. I wrote, "A cat is not a rigid body and is absolutely accelerating towards Earth, experiencing the physical effects of gravity." Are you saying that is wrong?
Consider this: if you hold an accelerometer while stationary on the ground, it will read 1g (accelerating). If you read an accelerometer while in free-fall, it will read 0g (ignoring wind resistance). In the first scenario, you are accelerating compared to your rest frame, even if you are standing still.
This is not an arbitrary distinction either. The 1g of acceleration while stationary on the ground produces measurable relativistic effects.