This question here really interested me to think about what happens when radius decreases to a very small value, and how this contributes to an extreme increase in linear velocity. I'm not at all questioning the validity of angular momentum conservation, I'm just curious what I'm missing.
Let's say we have a point mass spinning on a string of radius $r_0$ away on a finger, like in many angular momentum demonstrations, with a velocity $v_0$. The initial angular momentum will be $mv_0r_0$. Let's say we bring the radius in really close, from, for example, 1 meter to .02 meters, by pulling down on the string to reduce the radius. This is a reduction of 50 times the original radius, which means that linear velocity must increase because moment of inertia decreases by a factor of 2500 ($M(R_0/50)^2)$ and therefore $\omega$ increases 2500 times its original value, and so $v = \omega / r$ increases to 50 times its original value. But if $v$ was originally, let's say $1m/s$, then going to $50m/s$ seems extremely unreasonable when the radius is only $2cm$. On top of that, the string should theoretically break due to the centripetal force required to keep it moving in the small circle, but this does not happen in a physics demonstration.
Also, the amount of work required to increase the velocity to 50 times is unreasonably high. There's something missing here. What is it exactly?