Why do we need front brakes?
I think you need to separate operator error from optimal mechanical functioning.  

Mechanical advantage

By your own anecdotal evidence you have demonstrated how powerful front brakes can be.  In short we have front brakes because they are the most powerful brake. When a bike (or any vehicle) decelerates weight is shifted to the front wheel.  Because bicycle tires work through static friction, and static friction is proportional to the force applied, under braking conditions the front wheel will have the most traction (assuming no additional forces, such as turning are applied).  By contrast the harder you brake the less force is applied to the rear wheel and the static friction of the rear tire decreases. As a result the harder you brake, the less traction the rear wheel has, which reduces maximal stopping force (a bit of a negative feedback loop going on here).

The problem as you identify is not with the physics behind the inclusion of the front brake but operator error.

Correct Operation

As your anecdotal stories indicate that many bicycle operators are unfamiliar proper body position during emergency braking. If you learn to move your weight back (ideally behind your saddle) during strong braking then going over the handle bars is nearly impossible (except under very steep hills).  On the flat, with correct body position, it is possible to break traction with the front wheel before going over the handle bars.

The issue then becomes one of education.  In essence you are asking whether front brakes should be removed because sufficient education on how to operate a bicycle is lacking for many.  That is a political question and is unclear if a "right" answer exists.

Physics fun

Event though a wheel is rolling traction is governed by the laws of static friction. When part of the tire touches the ground, it is not sliding across the ground (usually). From a physics perspective this can be considered a body a "rest" – as strange as that sounds.

The resistive force observed in a body at rest  is defined as

Fs ≤ μ * Fn

Where Fs is the resistive force, μ is the coefficient of friction and Fn is the normal force.  The normal force is the force applied perpendicularly to the measured resistive force, on an incline the normal force is less than on a flat surface.  The resistive force (Fs) can range from zero to a maximum of μ * Fn.

The resistive force is what does the braking or turning.  If braking or cornering exceeds the resistive force your tire slides. The coefficient of friction is related to factors such as tire compound, tire pressure and road surface.  The act of braking will not change it (unless you exceed the resistive force).  

Under braking  Fn increases for the front wheel and decreases for the rear wheel due to the weight shift. As such the maximal braking force that can be applied increase for front wheel and decreases for the rear wheel the harder you brake.

As long as you are not throwing other variables in, such as rider weight distribution (i.e., too much weight up front), changing road conditions, and exceeding a tires maximal resistive force (doing so no the front will make you crash).  The front wheel is the best place to focus braking forces.