My thinking is this. The magnetic field of a permanent magnet is ultimately created by the spin of aligned electrons. But the spin is an intrinsic property of electrons: they aren't actually spinning.
Forget, only for a moment, about the spin, which is not spinning as you mentioned. Think about electrons as tiny magnets. The magnetic dipole is their intrinsic property (existing independently of external influences).
If I put two bar magnets next to each, with the south pole of one aligned to the (north pole of the) other, they attract. But there are no magnetic monopoles to "pull" on.
The force fields from two magnetic sources add up. In the language of line of forces, instead of closed loops through and around the one magnet
now the line of forces are going in loops through both magnetic and in between them and around them.
It was Faraday who introduced the line of forces:
Faraday usually discusses lines of force as chains of polarized particles in a dielectric, yet sometimes Faraday discusses them as having an existence all their own as in stretching across a vacuum.
It is a pity that the original sources are rarely used to understand today's ideas about physical processes. Thinking about the forces between two magnets as lines of tiny magnets, chained one to another, is an easy way to imagine how two magnets interact.
The attractive force must come from the Lorentz force law, which means that the magnet is pulling on charges (either electrons or protons), which must be moving in an orthogonal direction to both the magnetic field and the direction of force.
The Lorentz force describes the influence of a magnetic field on moving charges, which is not the case here, as you can easily prove. In any position the approach of two magnets can be stopped and the force between them doesn’t disappear.
Does this mean that the magnetic field created by spinning electrons in one magnet are pulling on "moving"/"orbiting" electrons in another magnet?
It means that when the magnetic dipoles of the involved electrons in a magnet oriented in the same direction, their magnetic fields are superimposed. The same happens with two magnets. During the approach, the aligned electrons in both magnets are also superimposed.
(Now, if you think it's important, think about the spin again.)