What was the first intuition that lead us to quantify energy with work instead of momentum? I know where the formula of work comes from and how we get it using integration. The work formula is interesting but it was counter-intuitive for me to choose it to quantify energy. Instead I was inclined to state that energy should be F.t 
Is it in order to better fit the concept of potential energy? 
Moreover with work and kinetic energy as a scale, another counter-intuitive concept  gave me trouble is that power (of an engine for instance) increases as time passes. If I use the same amount of fuel over time to produce a constant force on an object, whatever the velocity it gains, why would my instataneous power should be higher if I don't spend more energy in the process? I would have been inclined to use the constant jerk instead.
I know that momentum are conserved (I can't explain why) and that a part of kinetic energy is converted and lost into thermic form. But I still don't get what lead us on the right track. I can't just swallow it whithout knowing what it is.  
I don't want to define a new type of energy. I know we can't just choose as we wish. Momentum as energy scale to describe transmission of energy in all physics phenomena, would have been linearly proportional to velocity whereas kinetic energy as defined today is not linear to it. Therefore we have two really different function growth in respect to displacement and time, two different degree which means that no constant can connect them.
How come momentum is really conserved for two object.  Does it not fail to account for the heat  produced in a collision (known as coming from the extra amount out of the V² of the kinetic formula), the heat which can be seen as many little particles colliding, each having a momentum too, momentum created though conserved?!
 A: 
Is it in order to better fit the concept of potential energy?"

Naturally, if you wish to define a new "type" of energy, then it must still have units of Joules before we can call it energy. Otherwise, we would call it something else than energy. For example momentum.

[...] another counter-intuitive concept [that] gave me trouble is that power (of an engine for instance) increases as time passes.

Not true in all cases. Power is an amount of energy per second. The power of an engine (the power that the engine delivers to move the car) will rise as the engine starts and soon reaches a steady level. Fuel is then spent to provide the energy, and that will be at a constant rate of energy per second.

If I use the same amount of fuel over time to produce a constant force on an object, whatever the velocity it gains, why would my instataneous power should be higher if I don't spend more energy in the process?

I don't understand this question. Your instantaneous power might, as explained above, reach a constant level.

I know that momentum are conserved (I can't explain why) 

Momentum conservation can be derived from Newton's 3rd law. And this law is a law of nature which no one can prove. We have just seen it to be true infinitely many times, so we trust it to always be.

But I still don't get what lead us on the right track. I can't just swallow it whithout knowing what it is. 

The right track to what? This question is missing a word or two before it's a question :)
