Why is oil a better lubricant than water?

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.

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.

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.