Why does a particle moving on the inside of a cone not spiral towards the bottom?

The answer is the same reason the planets don't fall into the sun: as the particle gets closer to the center, its angular velocity goes up by angular momentum conservation. At a certain point, this costs so much energy that it's impossible to get any closer; we say the particle is 'repelled by the angular momentum barrier'. (For a real cone, friction would reduce the angular momentum over time, allowing the particle to spiral in.)

More quantitatively, the angular momentum is $$L = mv_\theta r$$ where $v_\theta$ is the tangential speed. Rearranging a little, we find that the kinetic energy due to this tangential motion is $$T_\theta = \frac{L^2}{2mr^2}.$$ That is, the energy cost of getting closer to the center diverges as $1/r^2$. In your situation, the potential energy only decreases linearly as we get closer to the center, $U \sim -r$, so eventually the angular momentum barrier will win and the particle can't get closer.

In the case of a planet orbiting the Sun, the potential energy diverges as $-1/r$, which still is beaten by the angular momentum barrier, since $1/r^2$ diverges faster. In general relativity, there is an additional term proportional to $-1/r^2$ in the potential, so things can fall into black holes.