How is the spin of hurricanes explained from an inertial frame?

Ok, the explanation that doesn't use the concept of fictitious force.

I need to mention first that explanation of physics taking place is independent of how you represent it. Example: a wind blows over a tree. When you see the fallen tree you know the cause: the wind blew it over. Whether you represent that in an inertial frame or a rotating frame has no bearing on identifying the cause.

The diagram below shows a schematic representation of the formation of cyclonic flow.

This representation is not to scale of course, the size is exaggerated for clarity.

The most interesting part of the diagram is the the air flow from east to west that is contributing to the formation of cyclonic flow. That east-to-west flow is curved so much that it moves inside of the latitude line that it is starting from.

To explain why east-to-west flowing air mass will move inside the latitude line it is starting from I need to discuss the way the rotating Earth is in an equilibrium shape.

the diagram shows a celestial body with a far larger equatorial bulge than the Earth.

Any buoyant object is subject to a normal force (red arrow).
When a celestial body has an equatorial bulge the gravitational force is not exactly opposite in direction to the normal force. So there is a resultant force, which in Geophysical Fluid Dynamics is referred to as the poleward force.

In the case of the Earth: polar radius is 6357 kilometer, equatorial radius is 6378 kilometer. The difference is 21 kilometer.

This means that from the equator to the pole is a downhill slope. If that slope would not be there then the water of the Earth would flow to the Equator. The Earth's equatorial bulge prevents that.

Example: at 45 degrees latitude the downward slope is 0.1 degree. That slope provides the required centripetal force to remain co-rotating with the Earth. At 45 degrees, to find the amount of required centripetal force you divide by 580, that is the ratio.

To get a feel for how much centripetal force is required (at 45 degrees latitude), divide your own weight by 580. If you have a weighing utensil you can push: that will give you a feel for it.

So: when you are at rest with respect to the Earth you are co-rotating with the Earth.

Now take the case of buoyant mass, flowing from west-to-east. Moving west-to-east the mass is circumnavigating the Earth's axis faster than the Earth itself. The mass is on the slope (like a car on a banked circuit), but the mass is speeding, so it will swing wide.
So: buoyant mass that is flowing from west-to-east will depart from the latitude that it is starting from, veering off towards the equator.

Now buoyant mass that is flowing from east-to-west.
Moving east-to-west this mass is circumnavigating the Earth's axis slower than the Earth itself. But the slope is the slope voor co-rotating mass. Comparison: if a car is on a banked circuit, and the car is going too slow the car will slump down

So that is what happens to air mass that is flowing from east-to-west: the poleward force pulls that air mass closer to the nearest pole.

For completeness:
When air mass is north-south, flowing away from the (nearest) pole the poleward force is doing negative work, slowing down the angular velocity of the air mass.

When air mass is south-north, flowing towards the (nearest) pole the poleward force is reeling in that mass, causing the angular velocity of the air mass to increase.

That is, highly schematicy, the process of formation of cyclonic flow.

Of course, that raises the question: why do many authors attribute the formation of cyclonic flow to a fictitious force?

To explain that let me discuss a simpler case: motion of a single object over the surface of a rotating celestial body.

The weight of the object is supported by the surface so the object is subject to the poleward force. At every latitude the magnitude of the poleward force is precisely the required centripetal for to keep co-rotating with the Earth, at the same distance to the axis of rotation.

(I'm also simplifying to planar motion, instead of three spatial dimenstions)

The formula for required centripetal force when circumnavigating at angular velocity $Omega$

$$ F = -m Omega^2 r $$

With the minus sign to indicate that this force acts back to the center, not outward.

The equation of motion for motion with respect to the rotating coordinate system has three terms:
the term for the poleward force
the coriolis term
the centrifugal term

$$ -m Omega^2 r + 2m Omega v + m Omega^2 r $$

The fact that the Earth's equatorial bulge is an equilibrium state is crucial here. The Earth's shape is such that at every latitude the poleward force provides the required centripetal force for co-rotating with the Earth. That is why the term for the poleward force has the angular velocity value $Omega$ of the Earth rotation.

That means that mathematically the term for the poleward force and the centrifugal term drop away against each other.

The equations of Geophysical Fluid Dynamics do incorporate the poleward force. If they would not incorporate the poleward force then the centrifugal term makes everything fly outward. The centrifugal term is far larger than the coriolis term. What happens is that the poleward force is incorporated and then it is the case that mathematically the term for the poleward force and the centrifugal term drop away against each other.