Why is the Long Line not a covering space for the Circle
The "obvious" function from the long line to the circle is not continuous at the limit ordinals. As you approach a limit ordinal from the left, the map revolves around the circle infinitely many times, and therefore does not converge to a limiting value on the circle.
Recall that every continuous map from the long line to the real numbers is eventually constant. This gives a contradiction with continuity of $p$ (as far as I can tell from a cursory glance at least).