This is a very smart question! Yes, reversibility does not have any intrinsic reference to time, but no, reversible processes are slow in practice. Let's talk about why.
We'll just start with the pebbles on a piston. What realistically happens if you pull the pebbles away too fast? Well, it's like making the pebble suddenly vanish: afterwards, the piston jumps up a tiny bit. When this happens, the piston immediately starts vibrating like a damped harmonic oscillator, and eventually the damping provided by the air and the walls of the piston leads to an energy transfer which increases entropy overall.
To minimize the vibration, you have to extract all of that would-be vibrational energy as work when you remove the pebble; this requires the forces to be applied slowly so that the piston is moving arbitrarily slowly when the pebble loses contact, so that the piston is left at rest at its new equilibrium point. In fact it should be moving arbitrarily slowly throughout the process to minimize friction losses as well.
Now what's the general principle here? The general principle is that we're reducing a source of pressure pushing down on the piston $P_0 \mapsto P_1$, and the resulting pressure gradient $P_1 - P_0$ between the gas and our pressing, causes the piston to move and the object to change volume with some $dV/dt.$ The gas loses energy at some rate $-P_0~dV/dt$ but we're harvesting energy at some rate $P_1~dV/dt$, so to recoup as much energy as possible we want $P_0 = P_1,$ or $P_1 - P_0 \to 0.$ That's what we want for reversibility.
Well if the pressure gradient is really driving the volume change then we'd expect something like $\frac{dV}{dt} = -\alpha~(P_1 - P_0),$ so reducing then reducing this difference means changing the volume over a very long time interval as $P_1 - P_0 \to 0.$
But the great thing about this is that it is such a generic argument. You might know that isothermal processes are reversible. What does this really mean? It means that you touch two objects at the same temperatures together, and let them exchange their thermal energy. "B.S.", you should be calling: "if they have the same temperature they shouldn't be able to trade internal energy." Very true. The above argument tells us that actually what we mean is the limit of a process where both objects have similar temperatures, but slightly different, so that they trade energy very very slowly. It has to be slow because $\frac{dE}{dt} \propto T_1 - T_0.$