Do hurricanes spin? It is often said that the spin of a hurricane forms as a result of being in a rotating reference frame with the spin caused by the coriolis force. For example, if you’re on a merry-go-round and you try to throw a ball forward, it will look like it’s curved from your perspective, but to the perspective of someone standing to the side, it looks like it goes straight forward.
If this is true, then why do hurricanes appear to rotate even for a satellite  observer in space? Shouldn’t they see the hurricane moving straight forward with no deflection?
 A: They do spin, but the reason for their spinning will differ depending on your frame of reference.  To an outside observer like the satellite you mentioned which sees the Earth rotating, the clouds on the North side of the storm try to drift Southward towards the eye of the storm due to the eye being lower pressure, but because the ground closer to the equator is moving Eastward faster than the Northern cloud is, the cloud falls behind and we see the northern part of the storm drift westward.
To a person standing on the Earth and rotating with the Earth, this behavior has no obvious cause and so we invent the Coriolis Force to account for it – i.e. we say that objects flying southwards will experience a westward deflection somewhat like gravity.  But this force does not exist in an inertial reference frame.
Note: maybe it's slightly more accurate to say the Coriolis Effect described above starts the hurricane spinning, and then it keeps spinning due to conservation of angular momentum, and continues to be influenced by various Coriolis and Centrifugal effects as well as obviously air currents.
A: You are framing your question in terms of 'what would a satellite observer in space see'.
I assume that what you are asking is:
To an observer who is not co-rotating with the Earth, how does air move as a hurricane starts to form'.
I propose to phrase the question as follows:
We define the following coordinate system: the non-rotating coordinate system that is co-moving with the center of mass of the Earth. In order to have a short name I will refer to this coordinate system as 'the inertial coordinate system' (To refer to it each time as the comoving-with-the-Earth-center-of-mass-inertial-coordinate-system is too many words, so I use the shortened 'the inertial coordinate system.)
We can have balloons suspended in the air; each balloons will at all times co-move with all of the air mass around it. We can plot the path of each balloon in the inertial coordinate system.
The interesting question then is:
As cyclonic flow starts to form, what does the path of the balloon with respect to the inertial coordinate system look like?



schematic representation of formation of cyclonic flow
Of course, in the image everything is enlarged, exaggerated, simplified. The image represents the fundamentals.

You mention the case of a trowing a ball from one part of a merry-go-round to another.
That is actually not a good comparison for the case of formation of cyclonic flow.
I will first discuss what is and is not a good comparison, and after that I will return to the formation of cyclonic flow.
There is a youtube video by Michael Stevens (Vsauce), that is largely about rotation. The title of the video is 'Laws and causes'
At 1:53 into that video Michael does the swivel-chair-and-heavy-objects-in-outstretched-arms demonstration
It being an educational video Michael Stevens uses heavy books for the demonstration.
The initial state is that an assistant has spun up the chair with Michael in it, Michael's arms are outstretched.
In that stage: the path of the books with respect to the inertial coordinate system (co-moving with the center of mass of Michael Stevens), is circumnavigating motion.
Then Michael pulls in his arms, and that causes his angular velocity to go up: Michael has increased his angular velocity all by himself.
The following diagram represents the path of the book with respect to the inertial coordinate sytem, and the force that Michael exerts is represented with a black arrow. That force is directed towards the axis of rotation.

circumnavigating object being pulled closer to axis of rotation
The path of the book with respect to the inertial coordinate system is along an inward spiral. We can think of that motion as a combination of circumnavigating motion and radial motion.
During that inward spiral Michael is exerting a surplus of centripetal force. (If Michael would extert precisely enough centripetal force then his effort would only sustain circumnavigating motion at uniform radial distance.) The effect of the surplus of centripetal force is that the book is pulled closer to the axis of rotation.
The grey arrows in the diagram represent a decomposition of the exerted force, in two perpendicular components; one component perpendicular to the instantaneous velocity, the other component parallel to the instantaneous velocity.
The force component parallel to the instantaneous velocity causes change of angular velocity
(In the video Michael gives the same force decomposition explanation, drawing a vividly colored diagram. That is at 10:00 into that video.)

Air mass
It is tempting to think of the air mass of the atmosphere as completely unconstrained.
But that is not actually the case; the air mass of the atmosphere is constrained to be co-rotating with the Earth, at a fairly constant distance to the Earth center of mass.
The simplest example is the air mass above the Equator, that air mass is in circumnavigating motion. The weight of that air mass is resting on the Earth. Air mass is not weightless,  it's not free floating mass: air mass is itself buoyant mass.
Air mass at latitudes other than the Equator remains co-rotating at the same latitude. We don't see all of the atmosphere sliding to the equator. A centripetal force is required. For example, at 45 degrees latitude the centripetal force that is required to maintain the circumnavigating motion at that latitude is the mass divided by 580.


The Merry-go-round and the ball
The case of throwing a ball from one part of a Merry-go-round to another part cannot meaningfully be used as example for formation of cyclonic flow. Once the ball is thrown its motion is no longer correlated with the motion of the Merry-go-round. Once the ball is thrown no force is acting upon it, and the motion of the ball with respect to the inertial coordinate system is along a straight line.
The meaningful example to use as basis of understanding the Earth rotation effect is the case of swivel-chair-and-pulling-in-heavy objects. The circumnavigating objects are never not subject to a force. At no point in time is the motion along a straight line.


Formation of cyclonic flow
Repeating the image I used at the start of this answer:

schematic representation of formation of cyclonic flow
At the start of this answer I restated the question as follows:
Follow balloons in their motion as they are co-moving with air mass. As cyclonic flow starts to form, what does the path of the balloon with respect to the inertial coordinate system look like?
The path of the balloon with respect to the inertial coordinate system is most definitely not along a straight line.

For more details:
Answer I posted in September 2020, to a question titled:
How is the spin of hurricanes explained from an inertial frame


(The images used in this answer are from an article about formation of cyclonic flow, that is my own website. A link to my own website is on my stackexchange profile page.)
