Intuitively, why is a reversible process one in which the system is always at equilibrium?

There are two ways to see this. A reversible process is the idealised limit of irreversible processes which go slower and slower. In the limit, a reversible process goes "infinitely slowly" (this is really a phrase used in some thermodynamics texts). I.e., it does not move at all. So the points have to be at equilibrium, or else they would be moving. When we call a process "reversible", we mean that it could go either way (but only if one disturbs the external conditions infintesimally, something which in fact alters the conditions under which the process is defined to be reversible, making it irreversible).But how could the system decide which way? It can't, of course, so in fact it does not move at all. It would only move if one altered the outside conditions a little bit, favouring one direction or the other, and thus making the process irreversible.

The other way to see this is by definition of an equilibrium point: an equilibrium point means that every real process which connects to that point must be leading in: none of them can be leading out. If a real, i.e., irreversible process started at a point and led away, the point would not be in equilibrium since that process would start going by Carnot's theorem.