Even in such a frame (for example, Painleve coordinates), not all of the statements are correct. In those coordinates, the infaller's coordinate acceleration is nonzero, as you say, but their coordinate speed inside the horizon is greater than the speed of light. Their inertial mass is still unchanged. And the coordinate acceleration in this frame is still not "towards" the singularity, since the singularity is still a moment of time, not a place in space.

> It's totally fine to say 'a astronaut in freefall is accelerating towards the earth', explicitly invoking Earth's reference frame.

In terms of coordinate acceleration, yes. But coordinate acceleration, precisely because it is frame-dependent, is not considered a physically meaningful quantity in relativity. Only invariants can be physically meaningful quantities. Proper acceleration--what an accelerometer attached to the infaller reads--is an invariant, and that is zero for a free-faller.

What happens but can never be observed, may aswell never happen from the p.o.v. of the observer. It has no consequences to say it does, or does not, happen (for if it did, those consequences would make it observable).

No, it isn't. The fact that a particular observer cannot detect an event does not in any way justify the claim that that event never happens.

> may aswell never happen from the p.o.v. of the observer

Whether or not an event "happens" is not a matter of any particular observer's point of view. At least, not in General Relativity. In GR, an event happening means there is a point in spacetime at which it happens. That point is either there in spacetime or it isn't, independent of what any observer can or cannot detect.

You can say an event has no causal effect on a particular observer, but that's not the same as saying the event never happens.