Well, physics is primarily an experimental science. We see something, we make a model that hopefully brings something new to check, we check, and either it is a good model or it is not.

Then come the theoreticians that try to find something deeper that would link or emerge the models.

In the case of energy conservation the "root" reason for the conservation are Noether principles that state some immutable entities (basically an experiment is not dependent on time and on geometric transformations such as translation or rotation). The link is not immediately visible (5 pages of math :)) but then suddenly tadaaam! and conservation of energy!

But yes, your point is very correct and this is why people wondered (and to some extend still wonder) if the masses in mhg=mv^2/2 are the same masses :)

Physics is really cool