Imagine voltage/potential at some point as density of electrons passing on a road. They all need space, so if there's a place near them with lower density, part of the electrons move there.
Now, let's say on the bottom end of the wire in question there is a potential, so the density of electrons is higher than usual. They see they can go up to the middle wire, and they do move there, but at some point they meet the electrons from the upper end of the wire, which has the same potential and same density of electrons. So, after they meet, they cannot go anymore, there's basically no direction they'd want to move to, because the density is now equal at all points, and therefore new electrons also no longer come from any end.
That way, we had a quick movement of eletrons which then established a balanced state, and no movement after that. But current is constant movement, so it doesn't count.
Constant movement can only be created by constant difference in density and we don't have that.
So, you had a right feeling that some movement from the ends to the middle part should happen, it's just that it lasts very short.
Also, you can imagine some gas moving through the pipes, with the same result. Imagine that batteries emit gas from one end (high potential) and suck it from the other end (low potential). That way you can clearly see how density of the gas smoothly decreases from high potential to low, therefore creating equal current across all the outer wire.