Why is oil a better lubricant than water? How come mineral oil is a better lubricant than water, even though water has a lower viscosity?

When two surfaces slide over each other with a gap filled with a fluid, the different layers of the fluid are dragged at different speeds. The very top layer touching the top metal surface will have the same speed as the surface itself, while the bottommost layer is stationary. The speed in the layers between is distributed linearly and there exist friction forces between those layers that slow the movement. Those frictional forces should be reduces however, if a fluid with a lower viscosity is chosen. 
How come this is not so? 
Does it have to do with water's polarity, so that it sticks to surfaces in a different way than oil?
 A: @tbf is right; lubrication, and tribology in general, is complicated. That's why there is that high effort to understand it and to design advanced materials.
There are several phenomena that cause the friction force exist and the ones you have neglected causes that oils are superior to water in most industral applications.
In dry sliding we can identify adhesion (dominant for two super-smooth glassy surfaces), asperity skipping and deformation (dominant for two rough and hard surfaces) and ploughing (dominant for sliding hard rough surface against soft one). Some add chemical bonding as separate cause, others consider it as a part of adhesion and another ones consider it as a condition.
The lubricants are chosen to lower the friction and wear and there is no universal superlubricant ideal for any application. One must consider:


*

*All materials in sliding contact;

*Range of the applied forces;

*Temperature;

*Sliding velocities;

*Environment (air/liquid flow, chemical surrounding, sliding frequency, debris presence, ...)


To the question, the mineral oil is good lubricant in case of sliding two metals because it passivates the surfaces and prevents their contact (adhesion is therefore neglected), if the viscosity is low enough it also decreses the interaction between asperities of both surfaces. On the other hand, water can chemically react with the surfaces and because of its low viscosity cannot prevent asperity interaction. But it says nothing in general.
Notes:
The most common lubricant on the Earth is water - joints in bodies of all vertebrates are water-lubed.
As Abhinav noted, graphite and all solid lubricants mentioned in comments below his answer are good lubricants and you cannot define viscosity there.
Turbomolecular pumps use magnetic bearings where "lubricant" is vacuum.
A: Water can not bear normal loads as well as oil.
Water is bound to escape from high pressure bearings to lower presser places in an open lubrication loop leaving bear contacts.
Water can create bubbles around cavities and corners and break the laminar flow which will compromise the separation of moving parts.
Water will react chemically with surfaces.
 There are lubricants mechanically designed to be near water viscosity but inert chemically and with wider temperature tolerance like brake fluids.
Many of the high speed revolving parts have been designed taking advantage of load bearing property of oil to actively and dynamically balance the system into its proper configuration under a range of different loadings or RPM, which is more practical with oil. Automatic transmission is but one case. 
A: A good lubricant tends to effectively minimize direct contact among components of any device that need it
Keeping this in mind, viscosity is not the only factor involved. Grind a graphite pencil lead, and it makes a mighty fine lubricant. It might be that in the case of water placed between two surfaces, a water droplet which was supposed to act as an intervening layer, gets displaced easily, resulting in untimely contact between the otherwise lubricated parts, resulting in wear and tear, while oil components tend to stay in place as the intervening medium and act as lubricant. Graphite obviously being a fine powder does not behave as water.
A: The parallel plate situation that you describe is not the typical condition encountered in practical lubrication operations.  In addition to facilitating the surfaces sliding over one another, the lubricated bearing must also support a normal load.  To do this, the gap between the surfaces varies with location along the bearing.  For example, in a journal bearing, the shaft will not be concentric with the bearing sleeve, and, in a slider bearing, the moving surface is at a a small angle to the stationary surface.  These features of the geometry allow pressure to build up in the gap between the surfaces as a result of a combination of drag flow and pressure flow.  This causes an upward normal load on the sliding member.  The higher the viscosity of the lubricant, the greater the pressure buildup and the greater the normal load that the bearing can support.  That's why we use lubricants with higher viscosity than water.
A: 
Why Oil is Slippery
Explaining why oil is slippery requires a look at its chemical
  properties. First, oil is non-polar, which means it does not have a
  positive or negative charge. Some molecules, like water, have a
  “charge distribution,” which means the molecule acts almost like a
  battery, part of it has a positive charge and part of it has a
  negative charge. The result, because positive is attracted to negative
  and vice versa, is that water and other “polar” molecules stick to
  each other. Oil doesn’t have this problem, so one oil molecule can
  slide past another more easily than one water molecule can slide past
  another.
Adding to the slipperiness of oil is its tendency to form distinct
  layers through forces called Van der Waals forces, or more
  specifically London Dispersion forces (a type of Van der Waals force).
  These forces, which are the weakest known in science, can help old
  things together, which would increase friction. However, oils have the
  unique property of forming forces only within layers because the
  molecules are essentially planar. Planar just means that molecules are
  flat as the diagram below emphasizes and only take up space in two
  dimensions rather than three. Without projections to attach to, forces
  can only be distributed within the plane and so there are no forces to
  bond one layer to the next. Thus, two layers of oil don’t bond to one
  another to any great degree.  ...



*

*http://www.petroleum.co.uk/oil-as-a-lubricant
I hope this answers your question

A: Your derivation is composed of correct statements and indeed, if something is known to act as a lubricant, we want the viscosity to be as low as possible because the friction will be reduced in this way. For example, honey is a bad lubricant because it's too viscous.
However, your derivation isn't the whole story. The second condition is that the two surfaces must stay apart. If you use a lubricant with too low a viscosity, the surfaces will come in contact and the original friction will reappear.
So the optimum lubricant is the least viscous liquid that is viscous enough to keep the surfaces apart. Which of them is the optimal one depends on the detailed surfaces and other conditions. For example, there exist situations in which water is a better lubricant than oil – for example when ice slides on ice. Some of the ice melts and the water is why the ice slides so nicely.
