Implications of Defining Work as Change in Mechanical Energy Would it be possible to form a theory of physics centered around the definition of work as the change in mechanical energy of an object/system, rather than the change in kinetic energy? What are some of the implications of using this definition? For instance, under this definition of work, conservative forces  cannot do work on the system which defines the potential function, since they only exchange potential energy for kinetic energy.
Is there a reason why we rather define work as just the change in kinetic energy? I am wondering if my alternative definition would be practical, since I personally think it is more intuitive.
 A: 
Would it be possible to form a theory of physics centered around the
definition of work as the change in mechanical energy of an
object/system, rather than the change in kinetic energy?

We don't "define" work this way. When we talk about work and change in kinetic energy we are referring to the work energy theorem which states:
The net work done on an object equals its change in kinetic energy.
Bold face is mine to emphasize that we are talking about net work and not just work. This is necessary because work can be negative or positive. If the net work is positive, the kinetic energy of the object increases, negative it decreases, and zero no change in kinetic energy.
But no change in kinetic energy does not necessarily mean no change in potential energy which is a system property and not a property of an object alone. For example, if we lift an object of mass $m$ an height from the ground $h$ starting from rest and ending at rest at $h$ the change in kinetic energy is zero. The positive work w do lifting the object equals the negative work done by gravity. The work done by gravity is negative because the direction of the force of gravity is opposite to the displacement of the object. Gravity takes the energy we gave the object and stores it as gravitational potential energy of the earth/object system.

For instance, under this definition of work, conservative forces
cannot do work on the system which defines the potential function,
since they only exchange potential energy for kinetic energy.

Conservative forces don't only exchange potential energy for kinetic energy. That only applies if the conservative force does positive work on an object.
In the example I gave the force of gravity (a conservative force) did negative work taking the energy that an external agent gave the object and storing it as gravitational potential energy. There is no exchange of potential energy for kinetic energy. That would occur if the object is released from the height $h$. Then the only (net) work done is positive work by gravity increasing the kinetic energy of the object at the expense of gravitational potential energy.
Hope this helps.
A: So after a bit of thinking and reading the helpful comments from other answers, I came to the following conclusions:
The net amount of work done to a system is not defined as the change in the system's kinetic energy. Instead, work done on an object/system from point a to point b is defined as the line integral $\int_a^bFds$, where F is the net force on the object or system. The Work Energy Theorem, which states that net work is equal to change in kinetic energy, follows mathematically from this definition (see this page for a 1-dimensional proof).
If we were to define net work as the change in mechanical energy of a system instead, then we would be in conflict with the line integral definition of work.
Consider a system which consists of an apple (1 kg) on the ground and the earth. If an external force raises the apple by 1 m and the apple ends at rest, the external force does +10 J of work. Gravity from the earth does -10 J of work on the apple, and gravity from the apple does ~0J of work on the earth. With all forces taken into account, we see that the net work done on the apple-earth system is 0 J.
One might think "let's instead define the net work as the change in mechanical energy, so that +10 J of net work was just done on the apple-earth system, instead of 0 J.  That seems to make more sense." But it actually doesn't make sense. As we see above, applying the definition of work to every object in the system gives us 0 J, not +10 J. If we were to implement the new definition of work, we would also have to change some other definitions (for example, the force-displacement definition of work would no longer apply. And that would be tragic.)
