An electron has no known internal structure, does that imply it has an unknown one?

Spin is not about stuff spinning. (Confusing, I know, but physicists have never been great at naming things. Exhibit A: Quarks.)

Spin is a purely quantum mechanical phenomenon, it cannot be understood with classical physics alone, and every analogy will break down. It has also, intrinsically, nothing to do with any kind of internal structure.

(Non-relativistic) spin arises simply because quantum things must transform in some representation of the rotation group $\mathrm{SO}(3)$ in order for the operators of angular momentum to act upon them (and because we need to explain the degree of freedom observed in, e.g., the Stern-Gerlach experiment. Since the states in the QM space of states are only determined up to rays, we seek a projective representation upon the space, and this means that we actually represent the covering group $\mathrm{SU}(2)$. The $\mathrm{SU}(2)$ representations are labeled by a number $s \in \mathbb{N} \vee s \in \mathbb{N} + \frac{1}{2}$, which we call spin. Whether the thing we are looking at is "composite" or "fundamental" has no impact on the general form of this argument.

"The electron has no known internal structure", but since it does have a spin, does that mean that we know the electron has an internal structure but we just don't know what it is?

An electron has no known internal structure simply means that nobody knows if the electron has an internal structure. So far they know none and therefore they suppose it has none

Spin is not related to an internal structure. If you consider the electron as the classical ball, the ball can spin both with or without an internal structure.

But spin is now considered an intrinsic property of the electron, which means that the effects are those of a classical spin, but the particle must not necessarily spin.

Spin is a wave property. It exists in classical relativistic wave theories as well. A circularly polarized wave carries an angular momentum that's related to the spin of the field. A gravitational wave (spin-2) can carry twice the angular momentum of a classical electromagnetic wave (spin-1).

Being "pointlike" is a particle property. You can think of the field value at a point as being related to the presence of a particle there. If the field's associated particle is an extended object (like a pion) then it doesn't just occupy the point where the field is nonzero, but also nearby points, which means that the interaction with a pointlike test particle depends not only on the field value at the test particle's location, but also on nearby field values. If the field's particle is pointlike (like the photon) then the force depends only on the field value at that point. Even classically, you could say that the electromagnetic field is "pointlike" since the Lorentz force only depends on the field at a point, though the terminology makes less sense without wave-particle duality.

So while spin is certainly a property of the particle (inasmuch as the particle and the wave are the same thing), it's not a property that depends on any internal structure of the particle.