First off: recommendation of the following two resources:
2018 Article by Nicholas Mecholsky Analytic formula for the geometric phase of an Asymmetric top
2020 Youtube video by David Brown The Dzhanibekov effect, equations and simulations
The video by David Brown is particularly clarifying.
David Brown has set up simulation of the most symmetric case where there are still three different moments of inertia:
Screenshot from the first video (of 2 videos) about the intermediate axis theorem
The implementation of the simulation makes the following clear: when the object is rotating there are internal stresses. If the struts would be flexible they would flex, dissipating kinetic energy. (In the equations for the simulation the object is for simplicity treated as perfectly rigid, of course.)
These internal forces are continuously relocating momentum from one part of the asymmetric top to another. The orientation and magnitude of the global angular momentum is constant (since there is no external torque). The continuous internal relocation of momentum gives the pattern of continuous change of the orientation of the angular velocity vector with respect to the asymmetric top.
David Brown points that there are only two circumstances where (internal) relocation of momentum does not occur: when the rotation is exactly along the axis of largest moment of inertia, and when the rotation is exactly along the axis of smallest moment of inertia.
In all other circumstances internal relocation of momentum does occur. That is: when the rotation is just a little bit off the axis of largest/smallest moment of inertia there is continuous relocation of momentum too, it's just less vivid.
Pendulum with a rigid rod
David Brown offers the following comparison: a pendulum with a rigid rod. When the pendulum amplitude is very small the motion is to a close approximation harmonic oscillation. The motion pattern of an asymetric top that is rotating just slightly off the axis of smallest/largest moment of inertia looks like just a slight wobble, analogous to a pendulum swinging with small amplitude.
The other end of the spectrum is that the pendulum is released from a close to completely inverted position. Let me refer to that as being released at 175 degrees away from hanging vertically down. The pendulum will then swing back and fort between 175 degrees and minus 175 degrees. Visually that looks as if the pendulum is lingering in the inverted orientation, with an apparently sudden swing to the other inverted orientation.
Being released from close to completely inverted is a release with the potential energy of the pendulum maxed out. (As in: you cannot release with more inital potential energy than that.)
In the case of the asymmetric top:
Setting up the initial rotation state very close to the axis of intermediate moment of inertia is like the case of releasing an inverted pendulum from close to completely inverted.
Dissipation of kinetic energy
I think it is also very instructive to consider what happens when the system does have significant dissipation of kinetic energy. (Here I mean with dissipation of kinetic energy: internal dissipation due to flexing of the structure, not dissipation due to friction from something external.)
Angular velocity around the axis of largest moment of inertia has the masses furthest away from the center of rotation. So: for the same angular momentum the rotation rate will be the slowest.
In the presence of dissipation of kinetic energy:
If the initial state of rotation is that the angular velocity vector is close to the axis of intermediate moment of inertia then the asymmetric top will over time proceed to a state where the rotation is along the axis of largest moment of inertia.
The initial state of rotation being close to the intermediate moment of inertia is in a sense a maxed out state.
There is an end state such that all the kinetic energy that can dissipate has dissipated. That end state is rotation around the axis of largest moment of inertia. (There is still kinetic energy at that point, but it has no opportunity to dissipate.)
An initial state of rotation close to the axis of intermedite moment of inertia is a state that is the most charged with additional kinetic energy, as compared to state of rotation around the axis of largest moment of inertia.