How does spinning help stabilize a ball?
Why does spinning help stabilize?
This answer answers the title "Why does spinning help stabilize?" And not specifically for takeoffs.
Before you jump into the answer, you should take a look at the background. For rigid bodies, adding a spin helps compensate for asymmetries. Neither missiles nor satellites are rigid, they contain liquids and flexible attachments, and both are means of energy attenuation and means of transferring energy from one axis of rotation to another.
The basic ideas of stability in this context (stability is a big issue, even in math, before trying to apply it to a real object) is that objects that rotate can be stable when they revolve around their axes at their maximum or rotate with minimal inertia. Good examples would be the central axis of a coin (like a windmill) or a pencil (like a grill).
This does not apply to bodies with modes of energy transfer through propellant sloshing and attachments. A pencil-like rocket that rotates lengthways can, through flexible modes, transfer all of its rotational energy into rotation around its axis of maximum inertia. This means that an "end over end" movement is assumed. Please note that I am not suggesting that the angular momentum has changed here. There is a case where this really happens with an early small launcher (late 50's I believe).
If there are flexible modes, the only stable rotation around the axis of maximum inertia is windmill rotation instead of grill mode.
Spin stabilization is used in many satellite designs, although in each case the developers must pay close attention to the shape of the satellite so that it behaves firmly in windmill mode, that is, rotates around its axis with maximum inertia.
Additional detail (EDIT)
The early small launch vehicle I mentioned was the Explorer 1 / Juno 1 mission, the first US satellite. It was supposed to rotate at 750 RPM along the longitudinal axis, but it was found to rotate at 7.5-8 RPM, which is the amount of energy transferred into one of the transverse axes. It's famous enough to be an example of classroom dynamics. This picture below shows the whip antennas which I believe are the physical mechanism for transferring energy between the axes.
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