Both you and the article are right - you are analyzing it in the frame of reference where the center of the earth is stationary, while the article analyzes in the frame of motion where the center of the rocket is stationary.
Since the frame where the center of the rocket is stationary is a non-inertial frame, Newton's law doesn't apply [1]. However a modification of Newton's law that includes a so-called "ficticious force" applies [2] (I don't think this modification has a name). This is why the article says there's an outward acceleration, because in the frame where the center of the rocket is stationary, the outward acceleration is caused by the ficticious force.
[2]: https://en.wikipedia.org/wiki/Non-inertial_reference_frame quotes, "One might say that F = ma holds in any coordinate system provided the term 'force' is redefined to include the so-called 'reversed effective forces' or 'inertia forces'."
Right, of course I understand that - but two points.
1) The whole original post appears to be worded for the layman to try and dispel myths around what orbit is. I don't think the layman is going to think about inertial frames of reference, rather I genuinely think the common misconception that there's really something 'pulling' the astronauts up will continue, much like many people think a car is throwing them out of a bend in the road, rather than the vehicle pushing them away from a straight line.
2) In any case, even if Musk implies the frame of reference, there's a reason a 'centrifugal force' is also referred to as a 'fictitious force' [1]
Since the frame where the center of the rocket is stationary is a non-inertial frame, Newton's law doesn't apply [1]. However a modification of Newton's law that includes a so-called "ficticious force" applies [2] (I don't think this modification has a name). This is why the article says there's an outward acceleration, because in the frame where the center of the rocket is stationary, the outward acceleration is caused by the ficticious force.
[1]: https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion begins the laws with "when viewed in an inertial reference frame"
[2]: https://en.wikipedia.org/wiki/Non-inertial_reference_frame quotes, "One might say that F = ma holds in any coordinate system provided the term 'force' is redefined to include the so-called 'reversed effective forces' or 'inertia forces'."